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# -*- coding: utf-8 -*- """ Binary Trees
This module deals with binary trees as mathematical (in particular immutable) objects.
.. NOTE::
If you need the data-structure for example to represent sets or hash tables with AVL trees, you should have a look at :mod:`sage.misc.sagex_ds`.
AUTHORS:
- Florent Hivert (2010-2011): initial implementation. - Adrien Boussicault (2015): Hook statistics.
REFERENCES:
.. [LodayRonco] Jean-Louis Loday and Maria O. Ronco. *Hopf algebra of the planar binary trees*, Advances in Mathematics, volume 139, issue 2, 10 November 1998, pp. 293-309. http://www.sciencedirect.com/science/article/pii/S0001870898917595
.. [HNT05] Florent Hivert, Jean-Christophe Novelli, and Jean-Yves Thibon. *The algebra of binary search trees*, :arxiv:`math/0401089v2`.
.. [CP12] Grégory Châtel, Viviane Pons. *Counting smaller trees in the Tamari order*, :arxiv:`1212.0751v1`. """ #***************************************************************************** # Copyright (C) 2010 Florent Hivert <Florent.Hivert@univ-rouen.fr>, # # Distributed under the terms of the GNU General Public License (GPL) # as published by the Free Software Foundation; either version 2 of # the License, or (at your option) any later version. # http://www.gnu.org/licenses/ #***************************************************************************** # python3 from __future__ import division, absolute_import from six import add_metaclass
from sage.structure.list_clone import ClonableArray from sage.combinat.abstract_tree import (AbstractClonableTree, AbstractLabelledClonableTree) from sage.combinat.ordered_tree import LabelledOrderedTrees from sage.rings.integer import Integer from sage.misc.inherit_comparison import InheritComparisonClasscallMetaclass from sage.misc.lazy_attribute import lazy_class_attribute from sage.combinat.combinatorial_map import combinatorial_map from sage.structure.parent import Parent from sage.structure.unique_representation import UniqueRepresentation from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
from sage.sets.non_negative_integers import NonNegativeIntegers from sage.sets.disjoint_union_enumerated_sets import DisjointUnionEnumeratedSets from sage.sets.family import Family from sage.misc.cachefunc import cached_method from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
@add_metaclass(InheritComparisonClasscallMetaclass) class BinaryTree(AbstractClonableTree, ClonableArray): """ Binary trees.
Binary trees here mean ordered (a.k.a. plane) finite binary trees, where "ordered" means that the children of each node are ordered.
Binary trees contain nodes and leaves, where each node has two children while each leaf has no children. The number of leaves of a binary tree always equals the number of nodes plus `1`.
INPUT:
- ``children`` -- ``None`` (default) or a list, tuple or iterable of length `2` of binary trees or convertible objects. This corresponds to the standard recursive definition of a binary tree as either a leaf or a pair of binary trees. Syntactic sugar allows leaving out all but the outermost calls of the ``BinaryTree()`` constructor, so that, e. g., ``BinaryTree([BinaryTree(None),BinaryTree(None)])`` can be shortened to ``BinaryTree([None,None])``. It is also allowed to abbreviate ``[None, None]`` by ``[]``.
- ``check`` -- (default: ``True``) whether check for binary should be performed or not.
EXAMPLES::
sage: BinaryTree() . sage: BinaryTree(None) . sage: BinaryTree([]) [., .] sage: BinaryTree([None, None]) [., .] sage: BinaryTree([None, []]) [., [., .]] sage: BinaryTree([[], None]) [[., .], .] sage: BinaryTree("[[], .]") [[., .], .] sage: BinaryTree([None, BinaryTree([None, None])]) [., [., .]]
sage: BinaryTree([[], None, []]) Traceback (most recent call last): ... ValueError: this is not a binary tree
TESTS::
sage: t1 = BinaryTree([[None, [[],[[], None]]],[[],[]]]) sage: t2 = BinaryTree([[[],[]],[]]) sage: with t1.clone() as t1c: ....: t1c[1,1,1] = t2 sage: t1 == t1c False """ @staticmethod def __classcall_private__(cls, *args, **opts): """ Ensure that binary trees created by the enumerated sets and directly are the same and that they are instances of :class:`BinaryTree`.
TESTS::
sage: from sage.combinat.binary_tree import BinaryTrees_all sage: issubclass(BinaryTrees_all().element_class, BinaryTree) True sage: t0 = BinaryTree([[],[[], None]]) sage: t0.parent() Binary trees sage: type(t0) <class 'sage.combinat.binary_tree.BinaryTrees_all_with_category.element_class'>
sage: t1 = BinaryTrees()([[],[[], None]]) sage: t1.parent() is t0.parent() True sage: type(t1) is type(t0) True
sage: t1 = BinaryTrees(4)([[],[[], None]]) sage: t1.parent() is t0.parent() True sage: type(t1) is type(t0) True """
@lazy_class_attribute def _auto_parent(cls): """ The automatic parent of the elements of this class.
When calling the constructor of an element of this class, one needs a parent. This class attribute specifies which parent is used.
EXAMPLES::
sage: BinaryTree._auto_parent Binary trees sage: BinaryTree([None, None]).parent() Binary trees """
def __init__(self, parent, children=None, check=True): """ TESTS::
sage: BinaryTree([None, None]).parent() Binary trees sage: BinaryTree("[., [., [., .]]]") [., [., [., .]]] sage: BinaryTree("[.,.,.]") Traceback (most recent call last): ... ValueError: this is not a binary tree sage: all(BinaryTree(repr(bt)) == bt for i in range(6) for bt in BinaryTrees(i)) True
`\QQ` (or any number field) has a ``list()`` method that returns itself as a `\QQ`-vector represented as a list. Before :trac:`23961`, this would cause an infinite recursion because `\QQ` elements give a list of length 1. For more details, see :trac:`23961`. We test that constructing binary trees from elements from `\QQ` terminates with an appropriate error::
sage: BinaryTree(1/2) Traceback (most recent call last): ... ValueError: this is not a binary tree """
children.parent() == parent): else: x.parent() == parent) else self.__class__(parent, x, check=check) for x in children]
def check(self): """ Check that ``self`` is a binary tree.
EXAMPLES::
sage: BinaryTree([[], []]) # indirect doctest [[., .], [., .]] sage: BinaryTree([[], [], []]) # indirect doctest Traceback (most recent call last): ... ValueError: this is not a binary tree sage: BinaryTree([[]]) # indirect doctest Traceback (most recent call last): ... ValueError: this is not a binary tree """ raise ValueError("this is not a binary tree")
def _repr_(self): """ TESTS::
sage: t1 = BinaryTree([[], None]); t1 # indirect doctest [[., .], .] sage: BinaryTree([[None, t1], None]) # indirect doctest [[., [[., .], .]], .] """ else:
def _ascii_art_(self): r""" TESTS::
sage: ascii_art(BinaryTree()) <BLANKLINE> sage: ascii_art(BinaryTree([])) o sage: for bt in BinaryTrees(3): ....: print(ascii_art(bt)) o \ o \ o o \ o / o o / \ o o o / o \ o o / o / o sage: ascii_art(BinaryTree([None,[]])) o \ o sage: ascii_art(BinaryTree([None,[None,[]]])) o \ o \ o sage: ascii_art(BinaryTree([None,[[],None]])) o \ o / o sage: ascii_art(BinaryTree([None,[[[],[]],[]]])) o \ _o_ / \ o o / \ o o sage: ascii_art(BinaryTree([None,[[None,[[],[]]],None]])) o \ o / o \ o / \ o o sage: ascii_art(BinaryTree([[],None])) o / o sage: ascii_art(BinaryTree([[[[],None], None],None])) o / o / o / o sage: ascii_art(BinaryTree([[[],[]],None])) o / o / \ o o sage: ascii_art(BinaryTree([[[None,[]],[[[],None],None]], None])) o / ___o___ / \ o o \ / o o / o sage: ascii_art(BinaryTree([[None,[[],[]]],None])) o / o \ o / \ o o sage: ascii_art(BinaryTree([[],[]])) o / \ o o sage: ascii_art(BinaryTree([[],[[],None]])) _o_ / \ o o / o sage: ascii_art(BinaryTree([[None,[]],[[[],None],None]])) ___o___ / \ o o \ / o o / o sage: ascii_art(BinaryTree([[[],[]],[[],None]])) __o__ / \ o o / \ / o o o sage: ascii_art(BinaryTree([[[],[]],[[],[]]])) __o__ / \ o o / \ / \ o o o o sage: ascii_art(BinaryTree([[[[],[]],[[],[]]],[]])) ___o___ / \ __o__ o / \ o o / \ / \ o o o o sage: ascii_art(BinaryTree([[],[[[[],[]],[[],[]]],[]]])) _____o______ / \ o ___o___ / \ __o__ o / \ o o / \ / \ o o o o """
else:
def _unicode_art_(self): r""" TESTS::
sage: unicode_art(BinaryTree()) <BLANKLINE> sage: unicode_art(BinaryTree([])) o sage: for bt in BinaryTrees(3): ....: print(unicode_art(bt)) o ╲ o ╲ o o ╲ o ╱ o o ╱ ╲ o o o ╱ o ╲ o o ╱ o ╱ o sage: unicode_art(BinaryTree([None,[]])) o ╲ o sage: unicode_art(BinaryTree([None,[None,[]]])) o ╲ o ╲ o sage: unicode_art(BinaryTree([None,[[],None]])) o ╲ o ╱ o sage: unicode_art(BinaryTree([None,[[[],[]],[]]])) o ╲ _o_ ╱ ╲ o o ╱ ╲ o o sage: unicode_art(BinaryTree([None,[[None,[[],[]]],None]])) o ╲ o ╱ o ╲ o ╱ ╲ o o sage: unicode_art(BinaryTree([[],None])) o ╱ o sage: unicode_art(BinaryTree([[[[],None], None],None])) o ╱ o ╱ o ╱ o sage: unicode_art(BinaryTree([[[],[]],None])) o ╱ o ╱ ╲ o o sage: unicode_art(BinaryTree([[[None,[]],[[[],None],None]], None])) o ╱ ___o___ ╱ ╲ o o ╲ ╱ o o ╱ o sage: unicode_art(BinaryTree([[None,[[],[]]],None])) o ╱ o ╲ o ╱ ╲ o o sage: unicode_art(BinaryTree([[],[]])) o ╱ ╲ o o sage: unicode_art(BinaryTree([[],[[],None]])) _o_ ╱ ╲ o o ╱ o sage: unicode_art(BinaryTree([[None,[]],[[[],None],None]])) ___o___ ╱ ╲ o o ╲ ╱ o o ╱ o sage: unicode_art(BinaryTree([[[],[]],[[],None]])) __o__ ╱ ╲ o o ╱ ╲ ╱ o o o sage: unicode_art(BinaryTree([[[],[]],[[],[]]])) __o__ ╱ ╲ o o ╱ ╲ ╱ ╲ o o o o sage: unicode_art(BinaryTree([[[[],[]],[[],[]]],[]])) ___o___ ╱ ╲ __o__ o ╱ ╲ o o ╱ ╲ ╱ ╲ o o o o sage: unicode_art(BinaryTree([[],[[[[],[]],[[],[]]],[]]])) _____o______ ╱ ╲ o ___o___ ╱ ╲ __o__ o ╱ ╲ o o ╱ ╲ ╱ ╲ o o o o """
else:
def _sort_key(self): """ Return a tuple of nonnegative integers encoding the binary tree ``self``.
The first entry of the tuple is the number of children of the root. Then the rest of the tuple is the concatenation of the tuples associated to these children (we view the children of a tree as trees themselves) from left to right.
This tuple characterizes the tree uniquely, and can be used to sort the binary trees.
EXAMPLES::
sage: x = BinaryTree([]) sage: y = (x.under(x)).over(x) sage: y._sort_key() (2, 2, 0, 0, 2, 0, 0) sage: z = (x.over(x)).under(x) sage: z._sort_key() (2, 2, 0, 2, 0, 0, 0) """
def is_empty(self): """ Return whether ``self`` is empty.
The notion of emptiness employed here is the one which defines a binary tree to be empty if its root is a leaf. There is precisely one empty binary tree.
EXAMPLES::
sage: BinaryTree().is_empty() True sage: BinaryTree([]).is_empty() False sage: BinaryTree([[], None]).is_empty() False """
def graph(self, with_leaves=True): """ Convert ``self`` to a digraph.
By default, this graph contains both nodes and leaves, hence is never empty. To obtain a graph which contains only the nodes, the ``with_leaves`` optional keyword variable has to be set to ``False``.
The resulting digraph is endowed with a combinatorial embedding, in order to be displayed correctly.
INPUT:
- ``with_leaves`` -- (default: ``True``) a Boolean, determining whether the resulting graph will be formed from the leaves and the nodes of ``self`` (if ``True``), or only from the nodes of ``self`` (if ``False``)
EXAMPLES::
sage: t1 = BinaryTree([[], None]) sage: t1.graph() Digraph on 5 vertices sage: t1.graph(with_leaves=False) Digraph on 2 vertices
sage: t1 = BinaryTree([[], [[], None]]) sage: t1.graph() Digraph on 9 vertices sage: t1.graph().edges() [(0, 1, None), (0, 4, None), (1, 2, None), (1, 3, None), (4, 5, None), (4, 8, None), (5, 6, None), (5, 7, None)] sage: t1.graph(with_leaves=False) Digraph on 4 vertices sage: t1.graph(with_leaves=False).edges() [(0, 1, None), (0, 2, None), (2, 3, None)]
sage: t1 = BinaryTree() sage: t1.graph() Digraph on 1 vertex sage: t1.graph(with_leaves=False) Digraph on 0 vertices
sage: BinaryTree([]).graph() Digraph on 3 vertices sage: BinaryTree([]).graph(with_leaves=False) Digraph on 1 vertex
sage: t1 = BinaryTree([[], [[], []]]) sage: t1.graph(with_leaves=False) Digraph on 5 vertices sage: t1.graph(with_leaves=False).edges() [(0, 1, None), (0, 2, None), (2, 3, None), (2, 4, None)] """
# Special treatment for the case when self is empty. # In this case, rec(self, 0) would give a false result.
# The edge set of res will be built up step by step using the # following function:
else: # tr is a node.
else: # We want only the nodes.
# Special treatment for the case when self has only 1 node. # In this case, the general DiGraph construction would # falsely yield an empty graph (since it adds nodes only # implicitly by adding edges).
# The edge set of res will be built up step by step using the # following function:
else: # tr is a node. else:
def canonical_labelling(self, shift=1): r""" Return a labelled version of ``self``.
The canonical labelling of a binary tree is a certain labelling of the nodes (not the leaves) of the tree. The actual canonical labelling is currently unspecified. However, it is guaranteed to have labels in `1...n` where `n` is the number of nodes of the tree. Moreover, two (unlabelled) trees compare as equal if and only if their canonical labelled trees compare as equal.
EXAMPLES::
sage: BinaryTree().canonical_labelling() . sage: BinaryTree([]).canonical_labelling() 1[., .] sage: BinaryTree([[[], [[], None]], [[], []]]).canonical_labelling() 5[2[1[., .], 4[3[., .], .]], 7[6[., .], 8[., .]]] """ self[1].canonical_labelling(shift + 1 + sz0)], label=shift + sz0) else:
def show(self, with_leaves=False): """ Show the binary tree ``show``, with or without leaves depending on the Boolean keyword variable ``with_leaves``.
.. WARNING::
For a labelled binary tree, the labels shown in the picture are not (in general) the ones given by the labelling!
Use :meth:`_latex_`, ``view``, :meth:`_ascii_art_` or ``pretty_print`` for more faithful representations of the data of the tree.
TESTS::
sage: t1 = BinaryTree([[], [[], None]]) sage: t1.show() """ except RuntimeError: # This is for the border case BinaryTree().show(). self.graph(with_leaves=with_leaves).show()
def make_node(self, child_list=[None, None]): """ Modify ``self`` so that it becomes a node with children ``child_list``.
INPUT:
- ``child_list`` -- a pair of binary trees (or objects convertible to)
.. NOTE:: ``self`` must be in a mutable state.
.. SEEALSO::
:meth:`make_leaf <sage.combinat.binary_tree.BinaryTree.make_leaf>`
EXAMPLES::
sage: t = BinaryTree() sage: t.make_node([None, None]) Traceback (most recent call last): ... ValueError: object is immutable; please change a copy instead. sage: with t.clone() as t1: ....: t1.make_node([None, None]) sage: t, t1 (., [., .]) sage: with t.clone() as t: ....: t.make_node([BinaryTree(), BinaryTree(), BinaryTree([])]) Traceback (most recent call last): ... ValueError: the list must have length 2 sage: with t1.clone() as t2: ....: t2.make_node([t1, t1]) sage: with t2.clone() as t3: ....: t3.make_node([t1, t2]) sage: t1, t2, t3 ([., .], [[., .], [., .]], [[., .], [[., .], [., .]]]) """
def make_leaf(self): """ Modify ``self`` so that it becomes a leaf (i. e., an empty tree).
.. NOTE:: ``self`` must be in a mutable state.
.. SEEALSO::
:meth:`make_node <sage.combinat.binary_tree.BinaryTree.make_node>`
EXAMPLES::
sage: t = BinaryTree([None, None]) sage: t.make_leaf() Traceback (most recent call last): ... ValueError: object is immutable; please change a copy instead. sage: with t.clone() as t1: ....: t1.make_leaf() sage: t, t1 ([., .], .) """
def _to_dyck_word_rec(self, usemap="1L0R"): r""" EXAMPLES::
sage: BinaryTree()._to_dyck_word_rec() [] sage: BinaryTree([])._to_dyck_word_rec() [1, 0] sage: BinaryTree([[[], [[], None]], [[], []]])._to_dyck_word_rec() [1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0] sage: BinaryTree([[None,[]],None])._to_dyck_word_rec("L1R0") [1, 1, 0, 0, 1, 0] sage: BinaryTree([[[], [[], None]], [[], []]])._to_dyck_word_rec("L1R0") [1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0] """ else:
@combinatorial_map(name="to the Tamari corresponding Dyck path") def to_dyck_word_tamari(self): r""" Return the Dyck word associated with ``self`` in consistency with the Tamari order on Dyck words and binary trees.
The bijection is defined recursively as follows:
- a leaf is associated with an empty Dyck word;
- a tree with children `l,r` is associated with the Dyck word `T(l) 1 T(r) 0`.
EXAMPLES::
sage: BinaryTree().to_dyck_word_tamari() [] sage: BinaryTree([]).to_dyck_word_tamari() [1, 0] sage: BinaryTree([[None,[]],None]).to_dyck_word_tamari() [1, 1, 0, 0, 1, 0] sage: BinaryTree([[[], [[], None]], [[], []]]).to_dyck_word_tamari() [1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0] """
def tamari_interval(self, other): r""" Return the Tamari interval between ``self`` and ``other`` as a :class:`~sage.combinat.interval_posets.TamariIntervalPoset`.
A "Tamari interval" is an interval in the Tamari poset. See :meth:`tamari_lequal` for the definition of the Tamari poset.
INPUT:
- ``other`` -- a binary tree greater or equal to ``self`` in the Tamari order
EXAMPLES::
sage: bt = BinaryTree([[None, [[], None]], None]) sage: ip = bt.tamari_interval(BinaryTree([None, [[None, []], None]])); ip The Tamari interval of size 4 induced by relations [(2, 4), (3, 4), (3, 1), (2, 1)] sage: ip.lower_binary_tree() [[., [[., .], .]], .] sage: ip.upper_binary_tree() [., [[., [., .]], .]] sage: ip.interval_cardinality() 4 sage: ip.number_of_tamari_inversions() 2 sage: list(ip.binary_trees()) [[., [[., [., .]], .]], [[., [., [., .]]], .], [., [[[., .], .], .]], [[., [[., .], .]], .]] sage: bt.tamari_interval(BinaryTree([[None,[]],[]])) Traceback (most recent call last): ... ValueError: The two binary trees are not comparable on the Tamari lattice.
TESTS:
Setting ``other`` equal to ``bt`` gives an interval consisting of just one element::
sage: ip = bt.tamari_interval(bt) sage: ip The Tamari interval of size 4 induced by relations [(1, 4), (2, 3), (3, 4), (3, 1), (2, 1)] sage: list(ip.binary_trees()) [[[., [[., .], .]], .]]
Empty trees work well::
sage: bt = BinaryTree() sage: ip = bt.tamari_interval(bt) sage: ip The Tamari interval of size 0 induced by relations [] sage: list(ip.binary_trees()) [.] """
def tamari_join(self, other): r""" Return the join of the binary trees ``self`` and ``other`` (of equal size) in the `n`-th Tamari poset (where `n` is the size of these trees).
The `n`-th Tamari poset (defined in :meth:`tamari_lequal`) is known to be a lattice, and the map from the `n`-th symmetric group `S_n` to the `n`-th Tamari poset defined by sending every permutation `p \in S_n` to the binary search tree of `p` (more precisely, to ``p.binary_search_tree_shape()``) is a lattice homomorphism. (See Theorem 6.2 in [Read04]_.)
.. SEEALSO::
:meth:`tamari_lequal`, :meth:`tamari_meet`.
AUTHORS:
Viviane Pons and Darij Grinberg, 18 June 2014; Frédéric Chapoton, 9 January 2018.
EXAMPLES::
sage: a = BinaryTree([None, [None, []]]) sage: b = BinaryTree([None, [[], None]]) sage: c = BinaryTree([[None, []], None]) sage: d = BinaryTree([[[], None], None]) sage: e = BinaryTree([[], []]) sage: a.tamari_join(c) == a True sage: b.tamari_join(c) == b True sage: c.tamari_join(e) == a True sage: d.tamari_join(e) == e True sage: e.tamari_join(b) == a True sage: e.tamari_join(a) == a True
::
sage: b1 = BinaryTree([None, [[[], None], None]]) sage: b2 = BinaryTree([[[], None], []]) sage: b1.tamari_join(b2) [., [[., .], [., .]]] sage: b3 = BinaryTree([[], [[], None]]) sage: b1.tamari_join(b3) [., [., [[., .], .]]] sage: b2.tamari_join(b3) [[., .], [., [., .]]]
The universal property of the meet operation is satisfied::
sage: def test_uni_join(p, q): ....: j = p.tamari_join(q) ....: if not p.tamari_lequal(j): ....: return False ....: if not q.tamari_lequal(j): ....: return False ....: for r in p.tamari_greater(): ....: if q.tamari_lequal(r) and not j.tamari_lequal(r): ....: return False ....: return True sage: all( test_uni_join(p, q) for p in BinaryTrees(3) for q in BinaryTrees(3) ) True sage: p = BinaryTrees(6).random_element(); q = BinaryTrees(6).random_element(); test_uni_join(p, q) True
Border cases::
sage: b = BinaryTree(None) sage: b.tamari_join(b) . sage: b = BinaryTree([]) sage: b.tamari_join(b) [., .]
REFERENCES:
.. [Read04] Nathan Reading. *Cambrian Lattices*. :arxiv:`math/0402086v2`. """ # We use Reading's result that the projection from the symmetric # group is a lattice homomorphism.
def tamari_meet(self, other, side="right"): r""" Return the meet of the binary trees ``self`` and ``other`` (of equal size) in the `n`-th Tamari poset (where `n` is the size of these trees).
The `n`-th Tamari poset (defined in :meth:`tamari_lequal`) is known to be a lattice, and the map from the `n`-th symmetric group `S_n` to the `n`-th Tamari poset defined by sending every permutation `p \in S_n` to the binary search tree of `p` (more precisely, to ``p.binary_search_tree_shape()``) is a lattice homomorphism. (See Theorem 6.2 in [Read04]_.)
.. SEEALSO::
:meth:`tamari_lequal`, :meth:`tamari_join`.
AUTHORS:
Viviane Pons and Darij Grinberg, 18 June 2014.
EXAMPLES::
sage: a = BinaryTree([None, [None, []]]) sage: b = BinaryTree([None, [[], None]]) sage: c = BinaryTree([[None, []], None]) sage: d = BinaryTree([[[], None], None]) sage: e = BinaryTree([[], []]) sage: a.tamari_meet(c) == c True sage: b.tamari_meet(c) == c True sage: c.tamari_meet(e) == d True sage: d.tamari_meet(e) == d True sage: e.tamari_meet(b) == d True sage: e.tamari_meet(a) == e True
::
sage: b1 = BinaryTree([None, [[[], None], None]]) sage: b2 = BinaryTree([[[], None], []]) sage: b1.tamari_meet(b2) [[[[., .], .], .], .] sage: b3 = BinaryTree([[], [[], None]]) sage: b1.tamari_meet(b3) [[[[., .], .], .], .] sage: b2.tamari_meet(b3) [[[[., .], .], .], .]
The universal property of the meet operation is satisfied::
sage: def test_uni_meet(p, q): ....: m = p.tamari_meet(q) ....: if not m.tamari_lequal(p): ....: return False ....: if not m.tamari_lequal(q): ....: return False ....: for r in p.tamari_smaller(): ....: if r.tamari_lequal(q) and not r.tamari_lequal(m): ....: return False ....: return True sage: all( test_uni_meet(p, q) for p in BinaryTrees(3) for q in BinaryTrees(3) ) True sage: p = BinaryTrees(6).random_element(); q = BinaryTrees(6).random_element(); test_uni_meet(p, q) True
Border cases::
sage: b = BinaryTree(None) sage: b.tamari_meet(b) . sage: b = BinaryTree([]) sage: b.tamari_meet(b) [., .] """
@combinatorial_map(name="to Dyck paths: up step, left tree, down step, right tree") def to_dyck_word(self, usemap="1L0R"): r""" Return the Dyck word associated with ``self`` using the given map.
INPUT:
- ``usemap`` -- a string, either ``1L0R``, ``1R0L``, ``L1R0``, ``R1L0``
The bijection is defined recursively as follows:
- a leaf is associated to the empty Dyck Word
- a tree with children `l,r` is associated with the Dyck word described by ``usemap`` where `L` and `R` are respectively the Dyck words associated with the trees `l` and `r`.
EXAMPLES::
sage: BinaryTree().to_dyck_word() [] sage: BinaryTree([]).to_dyck_word() [1, 0] sage: BinaryTree([[[], [[], None]], [[], []]]).to_dyck_word() [1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0] sage: BinaryTree([[None,[]],None]).to_dyck_word() [1, 1, 0, 1, 0, 0] sage: BinaryTree([[None,[]],None]).to_dyck_word("1R0L") [1, 0, 1, 1, 0, 0] sage: BinaryTree([[None,[]],None]).to_dyck_word("L1R0") [1, 1, 0, 0, 1, 0] sage: BinaryTree([[None,[]],None]).to_dyck_word("R1L0") [1, 1, 0, 1, 0, 0] sage: BinaryTree([[None,[]],None]).to_dyck_word("R10L") Traceback (most recent call last): ... ValueError: R10L is not a correct map
TESTS::
sage: bt = BinaryTree([[[], [[], None]], [[], []]]) sage: bt == bt.to_dyck_word().to_binary_tree() True sage: bt == bt.to_dyck_word("1R0L").to_binary_tree("1R0L") True sage: bt == bt.to_dyck_word("L1R0").to_binary_tree("L1R0") True sage: bt == bt.to_dyck_word("R1L0").to_binary_tree("R1L0") True """
def _to_ordered_tree(self, bijection="left", root=None): r""" Internal recursive method to obtain an ordered tree from a binary tree.
EXAMPLES::
sage: bt = BinaryTree([[],[]]) sage: bt._to_ordered_tree() [[], [[]]] sage: bt._to_ordered_tree(bijection="right") [[[]], []] sage: bt._to_ordered_tree(bijection="none") Traceback (most recent call last): ... ValueError: the bijection argument should be either left or right sage: bt = BinaryTree([[[], [[], None]], [[], []]]) sage: bt._to_ordered_tree() [[], [[], []], [[], [[]]]] sage: bt._to_ordered_tree(bijection="right") [[[[]], [[]]], [[]], []] """ root=None)) root=None)) root=root) else:
@combinatorial_map(name="To ordered tree, left child = left brother") def to_ordered_tree_left_branch(self): r""" Return an ordered tree of size `n+1` by the following recursive rule:
- if `x` is the left child of `y`, `x` becomes the left brother of `y` - if `x` is the right child of `y`, `x` becomes the last child of `y`
EXAMPLES::
sage: bt = BinaryTree([[],[]]) sage: bt.to_ordered_tree_left_branch() [[], [[]]] sage: bt = BinaryTree([[[], [[], None]], [[], []]]) sage: bt.to_ordered_tree_left_branch() [[], [[], []], [[], [[]]]] """
@combinatorial_map(name="To ordered tree, right child = right brother") def to_ordered_tree_right_branch(self): r""" Return an ordered tree of size `n+1` by the following recursive rule:
- if `x` is the right child of `y`, `x` becomes the right brother of `y` - if `x` is the left child of `y`, `x` becomes the first child of `y`
EXAMPLES::
sage: bt = BinaryTree([[],[]]) sage: bt.to_ordered_tree_right_branch() [[[]], []] sage: bt = BinaryTree([[[], [[], None]], [[], []]]) sage: bt.to_ordered_tree_right_branch() [[[[]], [[]]], [[]], []] """
def _postfix_word(self, left_first=True, start=1): r""" Internal recursive method to obtain a postfix canonical read of the binary tree.
EXAMPLES::
sage: bt = BinaryTree([[],[]]) sage: bt._postfix_word() [1, 3, 2] sage: bt._postfix_word(left_first=False) [3, 1, 2] sage: bt = BinaryTree([[[], [[], None]], [[], []]]) sage: bt._postfix_word() [1, 3, 4, 2, 6, 8, 7, 5] sage: bt._postfix_word(left_first=False) [8, 6, 7, 3, 4, 1, 2, 5] """ else:
def tamari_sorting_tuple(self): r""" Return the Tamari sorting tuple of ``self`` and the size of ``self``.
This is a pair `(w, n)`, where `n` is the number of nodes of ``self``, and `w` is an `n`-tuple whose `i`-th entry is the number of all nodes among the descendants of the right child of the `i`-th node of ``self``. Here, the nodes of ``self`` are numbered from left to right.
OUTPUT:
a pair `(w, n)`, where `w` is a tuple of integers, and `n` the size
Two binary trees of the same size are comparable in the Tamari order if and only if the associated tuples `w` are componentwise comparable. (This is essentially the Theorem in [HuangTamari1972]_.) This is used in :meth:`tamari_lequal`.
EXAMPLES::
sage: [t.tamari_sorting_tuple() for t in BinaryTrees(3)] [((2, 1, 0), 3), ((2, 0, 0), 3), ((0, 1, 0), 3), ((1, 0, 0), 3), ((0, 0, 0), 3)]
REFERENCES:
- [HuangTamari1972]_ """
@combinatorial_map(name="To 312 avoiding permutation") def to_312_avoiding_permutation(self): r""" Return a 312-avoiding permutation corresponding to the binary tree.
The linear extensions of a binary tree form an interval of the weak order called the sylvester class of the tree. This permutation is the minimal element of this sylvester class.
EXAMPLES::
sage: bt = BinaryTree([[],[]]) sage: bt.to_312_avoiding_permutation() [1, 3, 2] sage: bt = BinaryTree([[[], [[], None]], [[], []]]) sage: bt.to_312_avoiding_permutation() [1, 3, 4, 2, 6, 8, 7, 5]
TESTS::
sage: bt = BinaryTree([[],[]]) sage: bt == bt.to_312_avoiding_permutation().binary_search_tree_shape(left_to_right=False) True sage: bt = BinaryTree([[[], [[], None]], [[], []]]) sage: bt == bt.to_312_avoiding_permutation().binary_search_tree_shape(left_to_right=False) True """
@combinatorial_map(name="To complete tree") def as_ordered_tree(self, with_leaves=True): r""" Return the same tree seen as an ordered tree. By default, leaves are transformed into actual nodes, but this can be avoided by setting the optional variable ``with_leaves`` to ``False``.
EXAMPLES::
sage: bt = BinaryTree([]); bt [., .] sage: bt.as_ordered_tree() [[], []] sage: bt.as_ordered_tree(with_leaves = False) [] sage: bt = bt.canonical_labelling(); bt 1[., .] sage: bt.as_ordered_tree() 1[None[], None[]] sage: bt.as_ordered_tree(with_leaves=False) 1[] """ else: raise ValueError("The empty binary tree cannot be made into an ordered tree with with_leaves = False") else:
@combinatorial_map(name="To graph") def to_undirected_graph(self, with_leaves=False): r""" Return the undirected graph obtained from the tree nodes and edges.
Leaves are ignored by default, but one can set ``with_leaves`` to ``True`` to obtain the graph of the complete tree.
INPUT:
- ``with_leaves`` -- (default: ``False``) a Boolean, determining whether the resulting graph will be formed from the leaves and the nodes of ``self`` (if ``True``), or only from the nodes of ``self`` (if ``False``)
EXAMPLES::
sage: bt = BinaryTree([]) sage: bt.to_undirected_graph() Graph on 1 vertex sage: bt.to_undirected_graph(with_leaves=True) Graph on 3 vertices
sage: bt = BinaryTree() sage: bt.to_undirected_graph() Graph on 0 vertices sage: bt.to_undirected_graph(with_leaves=True) Graph on 1 vertex
If the tree is labelled, we use its labelling to label the graph. Otherwise, we use the graph canonical labelling which means that two different trees can have the same graph.
EXAMPLES::
sage: bt = BinaryTree([[],[None,[]]]) sage: bt.canonical_labelling().to_undirected_graph() == bt.to_undirected_graph() False sage: BinaryTree([[],[]]).to_undirected_graph() == BinaryTree([[[],None],None]).to_undirected_graph() True """ # this case needs extra care :(
def to_tilting(self): """ Transform a binary tree into a tilting object.
Let `t` be a binary tree with `n` nodes. There exists a unique depiction of `t` (above the diagonal) such that all leaves are regularly distributed on the diagonal line from `(0,0)` to `(n,n)` and all edges are either horizontal or vertical. This method provides the coordinates of this depiction, with the root as the top-left vertex.
OUTPUT:
a list of pairs of integers.
Every vertex of the binary tree is mapped to a pair of integers. The conventions are the following. The root has coordinates `(0, n)` where `n` is the node number. If a vertex is the left (right) son of another vertex, they share the first (second) coordinate.
EXAMPLES::
sage: t = BinaryTrees(1)[0] sage: t.to_tilting() [(0, 1)]
sage: for t in BinaryTrees(2): ....: print(t.to_tilting()) [(1, 2), (0, 2)] [(0, 1), (0, 2)]
sage: from sage.combinat.abstract_tree import from_hexacode sage: t = from_hexacode('2020222002000', BinaryTrees()) sage: print(t.to_tilting()) [(0, 1), (2, 3), (4, 5), (6, 7), (4, 7), (8, 9), (10, 11), (8, 11), (4, 11), (12, 13), (4, 13), (2, 13), (0, 13)]
sage: t2 = DyckWord([1,1,1,1,0,1,1,0,0,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0,0]).to_binary_tree() sage: len(t2.to_tilting()) == t2.node_number() True """ return []
def _to_tilting_rec(self, shift=0): """ Auxiliary method for :meth:`to_tilting`.
INPUT:
``shift`` -- an integer (default 0)
OUTPUT:
list of tilting coordinates and number of leaves
EXAMPLES::
sage: all(t._to_tilting_rec()[1] == 4 for t in BinaryTrees(3)) True """ else: else:
@combinatorial_map(name="To poset") def to_poset(self, with_leaves=False, root_to_leaf=False): r""" Return the poset obtained by interpreting the tree as a Hasse diagram.
The default orientation is from leaves to root but you can pass ``root_to_leaf=True`` to obtain the inverse orientation.
Leaves are ignored by default, but one can set ``with_leaves`` to ``True`` to obtain the poset of the complete tree.
INPUT:
- ``with_leaves`` -- (default: ``False``) a Boolean, determining whether the resulting poset will be formed from the leaves and the nodes of ``self`` (if ``True``), or only from the nodes of ``self`` (if ``False``) - ``root_to_leaf`` -- (default: ``False``) a Boolean, determining whether the poset orientation should be from root to leaves (if ``True``) or from leaves to root (if ``False``).
EXAMPLES::
sage: bt = BinaryTree([]) sage: bt.to_poset() Finite poset containing 1 elements sage: bt.to_poset(with_leaves=True) Finite poset containing 3 elements sage: P1 = bt.to_poset(with_leaves=True) sage: len(P1.maximal_elements()) 1 sage: len(P1.minimal_elements()) 2 sage: bt = BinaryTree([]) sage: P2 = bt.to_poset(with_leaves=True,root_to_leaf=True) sage: len(P2.maximal_elements()) 2 sage: len(P2.minimal_elements()) 1
If the tree is labelled, we use its labelling to label the poset. Otherwise, we use the poset canonical labelling::
sage: bt = BinaryTree([[],[None,[]]]).canonical_labelling() sage: bt 2[1[., .], 3[., 4[., .]]] sage: bt.to_poset().cover_relations() [[4, 3], [3, 2], [1, 2]]
Let us check that the empty binary tree is correctly handled::
sage: bt = BinaryTree() sage: bt.to_poset() Finite poset containing 0 elements sage: bt.to_poset(with_leaves=True) Finite poset containing 1 elements """ # this case needs extra care :(
@combinatorial_map(name="To 132 avoiding permutation") def to_132_avoiding_permutation(self): r""" Return a 132-avoiding permutation corresponding to the binary tree.
The linear extensions of a binary tree form an interval of the weak order called the sylvester class of the tree. This permutation is the maximal element of this sylvester class.
EXAMPLES::
sage: bt = BinaryTree([[],[]]) sage: bt.to_132_avoiding_permutation() [3, 1, 2] sage: bt = BinaryTree([[[], [[], None]], [[], []]]) sage: bt.to_132_avoiding_permutation() [8, 6, 7, 3, 4, 1, 2, 5]
TESTS::
sage: bt = BinaryTree([[],[]]) sage: bt == bt.to_132_avoiding_permutation().binary_search_tree_shape(left_to_right=False) True sage: bt = BinaryTree([[[], [[], None]], [[], []]]) sage: bt == bt.to_132_avoiding_permutation().binary_search_tree_shape(left_to_right=False) True """
def left_children_node_number(self, direction='left'): r""" Return the number of nodes which are left children in ``self``.
Every node (except the root) is either the left child or the right child of its parent node. The total number of nodes is `1` plus the number of left-children nodes plus the number of right-children nodes.
INPUT:
- ``direction`` -- either ``'left'`` (default) or ``'right'`` ; if set to ``'right'``, instead count nodes that are right children
EXAMPLES::
sage: bt = BinaryTree([[None,[[],[]]],[None,[[],None]]]) sage: ascii_art(bt) __o__ / \ o o \ \ o o / \ / o o o sage: bt.left_children_node_number('left') 3 sage: bt.left_children_node_number('right') 4
sage: all(5 == 1 + bt.left_children_node_number() ....: + bt.left_children_node_number('right') ....: for bt in BinaryTrees(5)) True
TESTS::
sage: BinaryTree([[],None]).left_children_node_number() 1 sage: BinaryTree([None,[]]).left_children_node_number() 0 sage: BinaryTree([]).left_children_node_number() 0 sage: BinaryTree().left_children_node_number() 0
sage: BinaryTree([[],None]).left_children_node_number('right') 0 sage: BinaryTree([None,[]]).left_children_node_number('right') 1 sage: BinaryTree([]).left_children_node_number('right') 0 sage: BinaryTree().left_children_node_number('right') 0 """
@combinatorial_map(order=2, name="Left-right symmetry") def left_right_symmetry(self): r""" Return the left-right symmetrized tree of ``self``.
EXAMPLES::
sage: BinaryTree().left_right_symmetry() . sage: BinaryTree([]).left_right_symmetry() [., .] sage: BinaryTree([[],None]).left_right_symmetry() [., [., .]] sage: BinaryTree([[None, []],None]).left_right_symmetry() [., [[., .], .]] """ return LabelledBinaryTree(tree, label=self.label())
@combinatorial_map(order=2, name="Left border symmetry") def left_border_symmetry(self): r""" Return the tree where a symmetry has been applied recursively on all left borders. If a tree is made of three trees `[T_1, T_2, T_3]` on its left border, it becomes `[T_3', T_2', T_1']` where same symmetry has been applied to `T_1, T_2, T_3`.
EXAMPLES::
sage: BinaryTree().left_border_symmetry() . sage: BinaryTree([]).left_border_symmetry() [., .] sage: BinaryTree([[None,[]],None]).left_border_symmetry() [[., .], [., .]] sage: BinaryTree([[None,[None,[]]],None]).left_border_symmetry() [[., .], [., [., .]]] sage: bt = BinaryTree([[None,[None,[]]],None]).canonical_labelling() sage: bt 4[1[., 2[., 3[., .]]], .] sage: bt.left_border_symmetry() 1[4[., .], 2[., 3[., .]]] """ else:
def canopee(self): """ Return the canopee of ``self``.
The *canopee* of a non-empty binary tree `T` with `n` internal nodes is the list `l` of `0` and `1` of length `n-1` obtained by going along the leaves of `T` from left to right except the two extremal ones, writing `0` if the leaf is a right leaf and `1` if the leaf is a left leaf.
EXAMPLES::
sage: BinaryTree([]).canopee() [] sage: BinaryTree([None, []]).canopee() [1] sage: BinaryTree([[], None]).canopee() [0] sage: BinaryTree([[], []]).canopee() [0, 1] sage: BinaryTree([[[], [[], None]], [[], []]]).canopee() [0, 1, 0, 0, 1, 0, 1]
The number of pairs `(t_1, t_2)` of binary trees of size `n` such that the canopee of `t_1` is the complementary of the canopee of `t_2` is also the number of Baxter permutations (see [DG94]_, see also :oeis:`A001181`). We check this in small cases::
sage: [len([(u,v) for u in BinaryTrees(n) for v in BinaryTrees(n) ....: if [1 - x for x in u.canopee()] == v.canopee()]) ....: for n in range(1, 5)] [1, 2, 6, 22]
Here is a less trivial implementation of this::
sage: from sage.sets.finite_set_map_cy import fibers sage: from sage.misc.all import attrcall sage: def baxter(n): ....: f = fibers(lambda t: tuple(t.canopee()), ....: BinaryTrees(n)) ....: return sum(len(f[i])*len(f[tuple(1-x for x in i)]) ....: for i in f) sage: [baxter(n) for n in range(1, 7)] [1, 2, 6, 22, 92, 422]
TESTS::
sage: t = BinaryTree().canopee() Traceback (most recent call last): ... ValueError: canopee is only defined for non empty binary trees
REFERENCES:
.. [DG94] \S. Dulucq and O. Guibert. Mots de piles, tableaux standards et permutations de Baxter, proceedings of Formal Power Series and Algebraic Combinatorics, 1994. """
else:
def in_order_traversal_iter(self): """ The depth-first infix-order traversal iterator for the binary tree ``self``.
This method iters each vertex (node and leaf alike) of the given binary tree following the depth-first infix order traversal algorithm.
The *depth-first infix order traversal algorithm* iterates through a binary tree as follows::
iterate through the left subtree (by the depth-first infix order traversal algorithm); yield the root; iterate through the right subtree (by the depth-first infix order traversal algorithm).
For example on the following binary tree `T`, where we denote leaves by `a, b, c, \ldots` and nodes by `1, 2, 3, \ldots`::
| ____3____ | | / \ | | 1 __7__ | | / \ / \ | | a 2 _5_ 8 | | / \ / \ / \ | | b c 4 6 h i | | / \ / \ | | d e f g |
the depth-first infix-order traversal algorithm iterates through the vertices of `T` in the following order: `a,1,b,2,c,3,d,4,e,5,f,6,g,7,h,8,i`.
See :meth:`in_order_traversal` for a version of this algorithm which not only iterates through, but actually does something at the vertices of tree.
TESTS::
sage: b = BinaryTree([[],[[],[]]]); ascii_art([b]) [ _o_ ] [ / \ ] [ o o ] [ / \ ] [ o o ] sage: ascii_art(list(b.in_order_traversal_iter())) [ , o, , _o_ , , o, , o , , o, ] [ / \ / \ ] [ o o o o ] [ / \ ] [ o o ] sage: ascii_art(filter(lambda node: node.label() is not None, ....: b.canonical_labelling().in_order_traversal_iter())) [ 1, _2_ , 3, 4 , 5 ] [ / \ / \ ] [ 1 4 3 5 ] [ / \ ] [ 3 5 ]
sage: list(BinaryTree(None).in_order_traversal_iter()) [.] """ # TODO:: PYTHON 3 # yield from self[0].in_order_traversal_iter() # TODO:: PYTHON 3 # yield from self[1].in_order_traversal_iter()
def in_order_traversal(self, node_action=None, leaf_action=None): r""" Explore the binary tree ``self`` using the depth-first infix-order traversal algorithm, executing the ``node_action`` function whenever traversing a node and executing the ``leaf_action`` function whenever traversing a leaf.
In more detail, what this method does to a tree `T` is the following::
if the root of `T` is a node: apply in_order_traversal to the left subtree of `T` (with the same node_action and leaf_action); apply node_action to the root of `T`; apply in_order_traversal to the right subtree of `T` (with the same node_action and leaf_action); else: apply leaf_action to the root of `T`.
For example on the following binary tree `T`, where we denote leaves by `a, b, c, \ldots` and nodes by `1, 2, 3, \ldots`::
| ____3____ | | / \ | | 1 __7__ | | / \ / \ | | a 2 _5_ 8 | | / \ / \ / \ | | b c 4 6 h i | | / \ / \ | | d e f g |
this method first applies ``leaf_action`` to `a`, then applies ``node_action`` to `1`, then ``leaf_action`` to `b`, then ``node_action`` to `2`, etc., with the vertices being traversed in the order `a,1,b,2,c,3,d,4,e,5,f,6,g,7,h,8,i`.
See :meth:`in_order_traversal_iter` for a version of this algorithm which only iterates through the vertices rather than applying any function to them.
INPUT:
- ``node_action`` -- (optional) a function which takes a node in input and does something during the exploration - ``leaf_action`` -- (optional) a function which takes a leaf in input and does something during the exploration
TESTS::
sage: nb_leaf = 0 sage: def l_action(_): ....: global nb_leaf ....: nb_leaf += 1 sage: nb_node = 0 sage: def n_action(_): ....: global nb_node ....: nb_node += 1
sage: BinaryTree().in_order_traversal(n_action, l_action) sage: nb_leaf, nb_node (1, 0)
sage: nb_leaf, nb_node = 0, 0 sage: b = BinaryTree([[],[[],[]]]); b [[., .], [[., .], [., .]]] sage: b.in_order_traversal(n_action, l_action) sage: nb_leaf, nb_node (6, 5) sage: nb_leaf, nb_node = 0, 0 sage: b = b.canonical_labelling() sage: b.in_order_traversal(n_action, l_action) sage: nb_leaf, nb_node (6, 5) sage: l = [] sage: b.in_order_traversal(lambda node: l.append( node.label() )) sage: l [1, 2, 3, 4, 5]
sage: leaf = 'a' sage: l = [] sage: def l_action(_): ....: global leaf, l ....: l.append(leaf) ....: leaf = chr( ord(leaf)+1 ) sage: n_action = lambda node: l.append( node.label() ) sage: b = BinaryTree([[None,[]],[[[],[]],[]]]).\ ....: canonical_labelling() sage: b.in_order_traversal(n_action, l_action) sage: l ['a', 1, 'b', 2, 'c', 3, 'd', 4, 'e', 5, 'f', 6, 'g', 7, 'h', 8, 'i'] """
def node_action(x): return None
else:
def tamari_lequal(self, t2): r""" Return ``True`` if ``self`` is less or equal to another binary tree ``t2`` (of the same size as ``self``) in the Tamari order.
The Tamari order on binary trees of size `n` is the partial order on the set of all binary trees of size `n` generated by the following requirement: If a binary tree `T'` is obtained by right rotation (see :meth:`right_rotate`) from a binary tree `T`, then `T < T'`. This not only is a well-defined partial order, but actually is a lattice structure on the set of binary trees of size `n`, and is a quotient of the weak order on the `n`-th symmetric group (also known as the right permutohedron order, see :meth:`~sage.combinat.permutation.Permutation.permutohedron_lequal`). See [CP12]_. The set of binary trees of size `n` equipped with the Tamari order is called the `n`-th Tamari poset.
The Tamari order can equivalently be defined as follows:
If `T` and `S` are two binary trees of size `n`, then the following four statements are equivalent:
- We have `T \leq S` in the Tamari order.
- There exist elements `t` and `s` of the Sylvester classes (:meth:`sylvester_class`) of `T` and `S`, respectively, such that `t \leq s` in the weak order on the symmetric group.
- The 132-avoiding permutation corresponding to `T` (see :meth:`to_132_avoiding_permutation`) is `\leq` to the 132-avoiding permutation corresponding to `S` in the weak order on the symmetric group.
- The 312-avoiding permutation corresponding to `T` (see :meth:`to_312_avoiding_permutation`) is `\leq` to the 312-avoiding permutation corresponding to `S` in the weak order on the symmetric group.
.. SEEALSO::
:meth:`tamari_smaller`, :meth:`tamari_greater`, :meth:`tamari_pred`, :meth:`tamari_succ`, :meth:`tamari_interval`
EXAMPLES:
This tree::
| o | | / \ | | o o | | / | | o | | / \ | | o o |
is Tamari-`\leq` to the following tree::
| _o_ | | / \ | | o o | | / \ \ | | o o o |
Checking this::
sage: b = BinaryTree([[[[], []], None], []]) sage: c = BinaryTree([[[],[]],[None,[]]]) sage: b.tamari_lequal(c) True
TESTS::
sage: for T in BinaryTrees(4): ....: for S in T.tamari_smaller(): ....: if S != T and T.tamari_lequal(S): ....: print("FAILURE") ....: if not S.tamari_lequal(T): ....: print("FAILURE") """ return False
def tamari_greater(self): r""" The list of all trees greater or equal to ``self`` in the Tamari order.
This is the order filter of the Tamari order generated by ``self``.
See :meth:`tamari_lequal` for the definition of the Tamari poset.
.. SEEALSO::
:meth:`tamari_smaller`
EXAMPLES:
For example, the tree::
| __o__ | | / \ | | o o | | / \ / | | o o o |
has these trees greater or equal to it::
|o , o , o , o , o , o ,| | \ \ \ \ \ \ | | o o o o _o_ __o__ | | \ \ \ \ / \ / \ | | o o o _o_ o o o o | | \ \ / \ / \ \ \ \ / | | o o o o o o o o o o | | \ \ \ / | | o o o o | | \ / | | o o | <BLANKLINE> | o , o , _o_ , _o__ , __o__ , ___o___ ,| | / \ / \ / \ / \ / \ / \ | | o o o o o o o _o_ o o o o | | \ \ / \ / \ \ \ \ / | | o o o o o o o o o o | | \ \ \ / \ \ | | o o o o o o | | \ / | | o o | <BLANKLINE> | _o_ , __o__ | | / \ / \ | | o o o o| | / \ \ / \ / | | o o o o o o |
TESTS::
sage: B = BinaryTree sage: b = B([None, B([None, B([None, B([])])])]);b [., [., [., [., .]]]] sage: b.tamari_greater() [[., [., [., [., .]]]]] sage: b = B([B([B([B([]), None]), None]), None]);b [[[[., .], .], .], .] sage: b.tamari_greater() [[., [., [., [., .]]]], [., [., [[., .], .]]], [., [[., .], [., .]]], [., [[., [., .]], .]], [., [[[., .], .], .]], [[., .], [., [., .]]], [[., .], [[., .], .]], [[., [., .]], [., .]], [[., [., [., .]]], .], [[., [[., .], .]], .], [[[., .], .], [., .]], [[[., .], [., .]], .], [[[., [., .]], .], .], [[[[., .], .], .], .]] """
def tamari_pred(self): r""" Compute the list of predecessors of ``self`` in the Tamari poset.
This list is computed by performing all left rotates possible on its nodes.
See :meth:`tamari_lequal` for the definition of the Tamari poset.
EXAMPLES:
For this tree::
| __o__ | | / \ | | o o | | / \ / | | o o o |
the list is::
| o , _o_ | | / / \ | | _o_ o o | | / \ / / | | o o o o | | / \ / | | o o o |
TESTS::
sage: B = BinaryTree sage: b = B([B([B([B([]), None]), None]), None]);b [[[[., .], .], .], .] sage: b.tamari_pred() [] sage: b = B([None, B([None, B([None, B([])])])]);b [., [., [., [., .]]]] sage: b.tamari_pred() [[[., .], [., [., .]]], [., [[., .], [., .]]], [., [., [[., .], .]]]] """ else: [B([g, s1], check=False) for g in s0.tamari_pred()] + [B([s0, d], check=False) for d in s1.tamari_pred()])
def tamari_smaller(self): r""" The list of all trees smaller or equal to ``self`` in the Tamari order.
This is the order ideal of the Tamari order generated by ``self``.
See :meth:`tamari_lequal` for the definition of the Tamari poset.
.. SEEALSO::
:meth:`tamari_greater`
EXAMPLES:
The tree::
| __o__ | | / \ | | o o | | / \ / | | o o o |
has these trees smaller or equal to it::
| __o__ , _o_ , o , o, o, o | | / \ / \ / / / / | | o o o o _o_ o o o | | / \ / / / / \ / \ / / | |o o o o o o o o o o o | | / / \ / / / | | o o o o o o | | / / \ / | | o o o o | | / | | o |
TESTS::
sage: B = BinaryTree sage: b = B([None, B([None, B([None, B([])])])]);b [., [., [., [., .]]]] sage: b.tamari_smaller() [[., [., [., [., .]]]], [., [., [[., .], .]]], [., [[., .], [., .]]], [., [[., [., .]], .]], [., [[[., .], .], .]], [[., .], [., [., .]]], [[., .], [[., .], .]], [[., [., .]], [., .]], [[., [., [., .]]], .], [[., [[., .], .]], .], [[[., .], .], [., .]], [[[., .], [., .]], .], [[[., [., .]], .], .], [[[[., .], .], .], .]] sage: b = B([B([B([B([]), None]), None]), None]);b [[[[., .], .], .], .] sage: b.tamari_smaller() [[[[[., .], .], .], .]] """
def tamari_succ(self): r""" Compute the list of successors of ``self`` in the Tamari poset.
This is the list of all trees obtained by a right rotate of one of its nodes.
See :meth:`tamari_lequal` for the definition of the Tamari poset.
EXAMPLES:
The list of successors of::
| __o__ | | / \ | | o o | | / \ / | | o o o |
is::
| _o__ , ___o___ , _o_ | | / \ / \ / \ | | o _o_ o o o o | | / \ \ / / \ \ | | o o o o o o o | | / \ | | o o |
TESTS::
sage: B = BinaryTree sage: b = B([B([B([B([]), None]), None]), None]);b [[[[., .], .], .], .] sage: b.tamari_succ() [[[[., .], .], [., .]], [[[., .], [., .]], .], [[[., [., .]], .], .]]
sage: b = B([]) sage: b.tamari_succ() []
sage: b = B([[],[]]) sage: b.tamari_succ() [[., [., [., .]]]] """ [B([g, self[1]]) for g in self[0].tamari_succ()] + [B([self[0], d]) for d in self[1].tamari_succ()])
def single_edge_cut_shapes(self): """ Return the list of possible single-edge cut shapes for the binary tree.
This is used in :meth:`sage.combinat.interval_posets.TamariIntervalPoset.is_new`.
OUTPUT:
a list of triples `(m, i, n)` of integers
This is a list running over all inner edges (i.e., edges joining two non-leaf vertices) of the binary tree. The removal of each inner edge defines two binary trees (connected components), the root-tree and the sub-tree. Thus, to every inner edge, we can assign three positive integers: `m` is the node number of the root-tree `R`, and `n` is the node number of the sub-tree `S`. The integer `i` is the index of the leaf of `R` on which `S` is grafted to obtain the original tree. The leaves of `R` are numbered starting from `1` (from left to right), hence `1 \leq i \leq m+1`.
In fact, each of `m` and `n` determines the other, as the total node number of `R` and `S` is the node number of ``self``.
EXAMPLES::
sage: BT = BinaryTrees(3) sage: [t.single_edge_cut_shapes() for t in BT] [[(2, 3, 1), (1, 2, 2)], [(2, 2, 1), (1, 2, 2)], [(2, 1, 1), (2, 3, 1)], [(2, 2, 1), (1, 1, 2)], [(2, 1, 1), (1, 1, 2)]]
sage: BT = BinaryTrees(2) sage: [t.single_edge_cut_shapes() for t in BT] [[(1, 2, 1)], [(1, 1, 1)]]
sage: BT = BinaryTrees(1) sage: [t.single_edge_cut_shapes() for t in BT] [[]] """ for m, i, n in left.single_edge_cut_shapes()] for m, i, n in right.single_edge_cut_shapes()]
def comb(self, side='left'): r""" Return the comb of a tree.
There are two combs in a binary tree: a left comb and a right comb.
Consider all the vertices of the leftmost (resp. rightmost) branch of the root. The left (resp. right) comb is the list of right (resp. left) subtrees of each of these vertices.
INPUT:
- ``side`` -- (default: 'left') set to 'left' to obtain a left comb, and to 'right' to obtain a right comb.
OUTPUT:
A list of binary trees.
.. SEEALSO::
:meth:`over_decomposition`, :meth:`under_decomposition`
EXAMPLES::
sage: BT = BinaryTree( '.' ) sage: [BT.comb('left'), BT.comb('right')] [[], []] sage: BT = BinaryTree( '[.,.]' ) sage: [BT.comb('left'), BT.comb('right')] [[], []] sage: BT = BinaryTree( '[[[.,.], .], [.,.]]' ) sage: BT.comb('left') [., .] sage: BT.comb('right') [.] sage: BT = BinaryTree( '[[[[., [., .]], .], [[., .], [[[., .], [., .]], [., .]]]], [., [[[., .], [[[., .], [., .]], .]], .]]]' ) sage: ascii_art(BT) ________o________ / \ __o__ o / \ \ o __o___ o / / \ / o o _o_ __o__ \ / \ / \ o o o o o / \ / o o o / \ o o sage: BT.comb('left') [[[., .], [[[., .], [., .]], [., .]]], ., [., .]] sage: ascii_art(BT.comb('left')) [ __o___ , , o ] [ / \ ] [ o _o_ ] [ / \ ] [ o o ] [ / \ ] [ o o ] sage: BT.comb('right') [., [[., .], [[[., .], [., .]], .]]] sage: ascii_art(BT.comb('right')) [ , __o__ ] [ / \ ] [ o o ] [ / ] [ o ] [ / \ ] [ o o ] """
def hook_number(self): r""" Return the number of hooks.
Recalling that a branch is a path from a vertex of the tree to a leaf, the leftmost (resp. rightmost) branch of a vertex `v` is the branch from `v` made only of left (resp. right) edges.
The hook of a vertex `v` is a set of vertices formed by the union of `{v}`, and the vertices of its leftmost and rightmost branches.
There is a unique way to partition the set of vertices in hooks. The number of hooks in such a partition is the hook number of the tree.
We can obtain this partition recursively by extracting the root's hook and iterating the processus on each tree of the remaining forest.
EXAMPLES::
sage: BT = BinaryTree( '.' ) sage: BT.hook_number() 0 sage: BT = BinaryTree( '[.,.]' ) sage: BT.hook_number() 1 sage: BT = BinaryTree( '[[[.,.], .], [.,.]]' ); ascii_art(BT) o / \ o o / o sage: BT.hook_number() 1 sage: BT = BinaryTree( '[[[[., [., .]], .], [[., .], [[[., .], [., .]], [., .]]]], [., [[[., .], [[[., .], [., .]], .]], .]]]' ) sage: ascii_art(BT) ________o________ / \ __o__ o / \ \ o __o___ o / / \ / o o _o_ __o__ \ / \ / \ o o o o o / \ / o o o / \ o o sage: BT.hook_number() 6 """ for t in self.comb('left') + self.comb('right'))
def twisting_number(self): r""" Return a pair (number of maximal left branches, number of maximal right branches).
Recalling that a branch of a vertex `v` is a path from a vertex of the tree to a leaf, a left (resp. right) branch is a branch made only of left (resp. right) edges. The length of a branch is the number of edges composing it. A left (resp. right) branch is maximal if it is not included in a strictly longer left (resp. right) branch.
OUTPUT :
A list of two integers.
EXAMPLES::
sage: BT = BinaryTree( '.' ) sage: BT.twisting_number() [0, 0] sage: BT = BinaryTree( '[.,.]' ) sage: BT.twisting_number() [0, 0] sage: BT = BinaryTree( '[[[.,.], .], [.,.]]' ); ascii_art(BT) o / \ o o / o sage: BT.twisting_number() [1, 1] sage: BT = BinaryTree( '[[[[., [., .]], .], [[., .], [[[., .], [., .]], [., .]]]], [., [[[., .], [[[., .], [., .]], .]], .]]]' ) sage: ascii_art(BT) ________o________ / \ __o__ o / \ \ o __o___ o / / \ / o o _o_ __o__ \ / \ / \ o o o o o / \ / o o o / \ o o sage: BT.twisting_number() [5, 6] sage: BT = BinaryTree( '[.,[[[.,.],.],.]]' ); ascii_art(BT) o \ o / o / o sage: BT.twisting_number() [1, 1] """
def q_hook_length_fraction(self, q=None, q_factor=False): r""" Compute the ``q``-hook length fraction of the binary tree ``self``, with an additional "q-factor" if desired.
If `T` is a (plane) binary tree and `q` is a polynomial indeterminate over some ring, then the `q`-hook length fraction `h_{q} (T)` of `T` is defined by
.. MATH::
h_{q} (T) = \frac{[\lvert T \rvert]_q!}{\prod_{t \in T} [\lvert T_t \rvert]_q},
where the product ranges over all nodes `t` of `T`, where `T_t` denotes the subtree of `T` consisting of `t` and its all descendants, and where for every tree `S`, we denote by `\lvert S \rvert` the number of nodes of `S`. While this definition only shows that `h_{q} (T)` is a rational function in `T`, it is in fact easy to show that `h_{q} (T)` is actually a polynomial in `T`, and thus makes sense when any element of a commutative ring is substituted for `q`. This can also be explicitly seen from the following recursive formula for `h_{q} (T)`:
.. MATH::
h_{q} (T) = \binom{ \lvert T \rvert - 1 }{ \lvert T_1 \rvert }_q h_{q} (T_1) h_{q} (T_2),
where `T` is any nonempty binary tree, and `T_1` and `T_2` are the two child trees of the root of `T`, and where `\binom{a}{b}_q` denotes a `q`-binomial coefficient.
A variation of the `q`-hook length fraction is the following "`q`-hook length fraction with `q`-factor":
.. MATH::
f_{q} (T) = h_{q} (T) \cdot \prod_{t \in T} q^{\lvert T_{\mathrm{right}(t)} \rvert},
where for every node `t`, we denote by `\mathrm{right}(t)` the right child of `t`. This `f_{q} (T)` differs from `h_{q} (T)` only in a multiplicative factor, which is a power of `q`.
When `q = 1`, both `f_{q} (T)` and `h_{q} (T)` equal the number of permutations whose binary search tree (see [HNT05]_ for the definition) is `T` (after dropping the labels). For example, there are `20` permutations which give a binary tree of the following shape::
| __o__ | | / \ | | o o | | / \ / | | o o o |
by the binary search insertion algorithm, in accordance with the fact that this tree satisfies `f_{1} (T) = 20`.
When `q` is considered as a polynomial indeterminate, `f_{q} (T)` is the generating function for all permutations whose binary search tree is `T` (after dropping the labels) with respect to the number of inversions (i. e., the Coxeter length) of the permutations.
Objects similar to `h_{q} (T)` also make sense for general ordered forests (rather than just binary trees), see e. g. [BW88]_, Theorem 9.1.
INPUT:
- ``q`` -- a ring element which is to be substituted as `q` into the `q`-hook length fraction (by default, this is set to be the indeterminate `q` in the polynomial ring `\ZZ[q]`)
- ``q_factor`` -- a Boolean (default: ``False``) which determines whether to compute `h_{q} (T)` or to compute `f_{q} (T)` (namely, `h_{q} (T)` is obtained when ``q_factor == False``, and `f_{q} (T)` is obtained when ``q_factor == True``)
REFERENCES:
.. [BW88] Anders Bjoerner, Michelle L. Wachs, *Generalized quotients in Coxeter groups*. Transactions of the American Mathematical Society, vol. 308, no. 1, July 1988. http://www.ams.org/journals/tran/1988-308-01/S0002-9947-1988-0946427-X/S0002-9947-1988-0946427-X.pdf
EXAMPLES:
Let us start with a simple example. Actually, let us start with the easiest possible example -- the binary tree with only one vertex (which is a leaf)::
sage: b = BinaryTree() sage: b.q_hook_length_fraction() 1 sage: b.q_hook_length_fraction(q_factor=True) 1
Nothing different for a tree with one node and two leaves::
sage: b = BinaryTree([]); b [., .] sage: b.q_hook_length_fraction() 1 sage: b.q_hook_length_fraction(q_factor=True) 1
Let us get to a more interesting tree::
sage: b = BinaryTree([[[],[]],[[],None]]); b [[[., .], [., .]], [[., .], .]] sage: b.q_hook_length_fraction()(q=1) 20 sage: b.q_hook_length_fraction() q^7 + 2*q^6 + 3*q^5 + 4*q^4 + 4*q^3 + 3*q^2 + 2*q + 1 sage: b.q_hook_length_fraction(q_factor=True) q^10 + 2*q^9 + 3*q^8 + 4*q^7 + 4*q^6 + 3*q^5 + 2*q^4 + q^3 sage: b.q_hook_length_fraction(q=2) 465 sage: b.q_hook_length_fraction(q=2, q_factor=True) 3720 sage: q = PolynomialRing(ZZ, 'q').gen() sage: b.q_hook_length_fraction(q=q**2) q^14 + 2*q^12 + 3*q^10 + 4*q^8 + 4*q^6 + 3*q^4 + 2*q^2 + 1
Let us check the fact that `f_{q} (T)` is the generating function for all permutations whose binary search tree is `T` (after dropping the labels) with respect to the number of inversions of the permutations::
sage: def q_hook_length_fraction_2(T): ....: P = PolynomialRing(ZZ, 'q') ....: q = P.gen() ....: res = P.zero() ....: for w in T.sylvester_class(): ....: res += q ** Permutation(w).length() ....: return res sage: def test_genfun(i): ....: return all( q_hook_length_fraction_2(T) ....: == T.q_hook_length_fraction(q_factor=True) ....: for T in BinaryTrees(i) ) sage: test_genfun(4) True """
else:
product_of_subtrees(b0) * product_of_subtrees(b1) * \ q ** (b1.node_number()) else: product_of_subtrees(b0) * product_of_subtrees(b1)
@combinatorial_map(name="Right rotate") def right_rotate(self): r""" Return the result of right rotation applied to the binary tree ``self``.
Right rotation on binary trees is defined as follows: Let `T` be a binary tree such that the left child of the root of `T` is a node. Let `C` be the right child of the root of `T`, and let `A` and `B` be the left and right children of the left child of the root of `T`. (Keep in mind that nodes of trees are identified with the subtrees consisting of their descendants.) Then, the right rotation of `T` is the binary tree in which the left child of the root is `A`, whereas the right child of the root is a node whose left and right children are `B` and `C`. In pictures::
| * * | | / \ / \ | | * C -right-rotate-> A * | | / \ / \ | | A B B C |
where asterisks signify a single node each (but `A`, `B` and `C` might be empty).
For example, ::
| o _o_ | | / / \ | | o -right-rotate-> o o | | / \ / | | o o o | <BLANKLINE> | __o__ _o__ | | / \ / \ | | o o -right-rotate-> o _o_ | | / \ / / \ | | o o o o o | | / \ \ | | o o o |
Right rotation is the inverse operation to left rotation (:meth:`left_rotate`).
The right rotation operation introduced here is the one defined in Definition 2.1 of [CP12]_.
.. SEEALSO::
:meth:`left_rotate`
EXAMPLES::
sage: b = BinaryTree([[[],[]], None]); ascii_art([b]) [ o ] [ / ] [ o ] [ / \ ] [ o o ] sage: ascii_art([b.right_rotate()]) [ _o_ ] [ / \ ] [ o o ] [ / ] [ o ] sage: b = BinaryTree([[[[],None],[None,[]]], []]); ascii_art([b]) [ __o__ ] [ / \ ] [ o o ] [ / \ ] [ o o ] [ / \ ] [ o o ] sage: ascii_art([b.right_rotate()]) [ _o__ ] [ / \ ] [ o _o_ ] [ / / \ ] [ o o o ] [ \ ] [ o ] """
@combinatorial_map(name="Left rotate") def left_rotate(self): r""" Return the result of left rotation applied to the binary tree ``self``.
Left rotation on binary trees is defined as follows: Let `T` be a binary tree such that the right child of the root of `T` is a node. Let `A` be the left child of the root of `T`, and let `B` and `C` be the left and right children of the right child of the root of `T`. (Keep in mind that nodes of trees are identified with the subtrees consisting of their descendants.) Then, the left rotation of `T` is the binary tree in which the right child of the root is `C`, whereas the left child of the root is a node whose left and right children are `A` and `B`. In pictures::
| * * | | / \ / \ | | A * -left-rotate-> * C | | / \ / \ | | B C A B |
where asterisks signify a single node each (but `A`, `B` and `C` might be empty).
For example, ::
| _o_ o | | / \ / | | o o -left-rotate-> o | | / / \ | | o o o | <BLANKLINE> | __o__ o | | / \ / | | o o -left-rotate-> o | | / \ / | | o o o | | / \ / \ | | o o o o | | / \ | | o o |
Left rotation is the inverse operation to right rotation (:meth:`right_rotate`).
.. SEEALSO::
:meth:`right_rotate`
EXAMPLES::
sage: b = BinaryTree([[],[[],None]]); ascii_art([b]) [ _o_ ] [ / \ ] [ o o ] [ / ] [ o ] sage: ascii_art([b.left_rotate()]) [ o ] [ / ] [ o ] [ / \ ] [ o o ] sage: b.left_rotate().right_rotate() == b True """
@combinatorial_map(name="Over operation on Binary Trees") def over(self, bt): r""" Return ``self`` over ``bt``, where "over" is the ``over`` (`/`) operation.
If `T` and `T'` are two binary trees, then `T` over `T'` (written `T / T'`) is defined as the tree obtained by grafting `T'` on the rightmost leaf of `T`. More precisely, `T / T'` is defined by identifying the root of the `T'` with the rightmost leaf of `T`. See section 4.5 of [HNT05]_.
If `T` is empty, then `T / T' = T'`.
The definition of this "over" operation goes back to Loday-Ronco [LodRon0102066]_ (Definition 2.2), but it is denoted by `\backslash` and called the "under" operation there. In fact, trees in sage have their root at the top, contrary to the trees in [LodRon0102066]_ which are growing upwards. For this reason, the names of the over and under operations are swapped, in order to keep a graphical meaning. (Our notation follows that of section 4.5 of [HNT05]_.)
.. SEEALSO::
:meth:`under`
EXAMPLES:
Showing only the nodes of a binary tree, here is an example for the over operation::
| o __o__ _o_ | | / \ / / \ = / \ | | o o o o o o | | \ / \ | | o o __o__ | | / \ | | o o | | \ / | | o o |
A Sage example::
sage: b1 = BinaryTree([[],[[],[]]]) sage: b2 = BinaryTree([[None, []],[]]) sage: ascii_art((b1, b2, b1/b2)) ( _o_ , _o_ , _o_ ) ( / \ / \ / \ ) ( o o o o o o_ ) ( / \ \ / \ ) ( o o o o o ) ( \ ) ( _o_ ) ( / \ ) ( o o ) ( \ ) ( o )
TESTS::
sage: b1 = BinaryTree([[],[]]); ascii_art([b1]) [ o ] [ / \ ] [ o o ] sage: b2 = BinaryTree([[None,[]],[[],None]]); ascii_art([b2]) [ __o__ ] [ / \ ] [ o o ] [ \ / ] [ o o ] sage: ascii_art([b1.over(b2)]) [ _o_ ] [ / \ ] [ o o ] [ \ ] [ __o__ ] [ / \ ] [ o o ] [ \ / ] [ o o ]
The same in the labelled case::
sage: b1 = b1.canonical_labelling() sage: b2 = b2.canonical_labelling() sage: ascii_art([b1.over(b2)]) [ _2_ ] [ / \ ] [ 1 3 ] [ \ ] [ __3__ ] [ / \ ] [ 1 5 ] [ \ / ] [ 2 4 ] """ else:
__div__ = over __truediv__ = over
@combinatorial_map(name="Under operation on Binary Trees") def under(self, bt): r""" Return ``self`` under ``bt``, where "under" is the ``under`` (`\backslash`) operation.
If `T` and `T'` are two binary trees, then `T` under `T'` (written `T \backslash T'`) is defined as the tree obtained by grafting `T` on the leftmost leaf of `T'`. More precisely, `T \backslash T'` is defined by identifying the root of `T` with the leftmost leaf of `T'`.
If `T'` is empty, then `T \backslash T' = T`.
The definition of this "under" operation goes back to Loday-Ronco [LodRon0102066]_ (Definition 2.2), but it is denoted by `/` and called the "over" operation there. In fact, trees in sage have their root at the top, contrary to the trees in [LodRon0102066]_ which are growing upwards. For this reason, the names of the over and under operations are swapped, in order to keep a graphical meaning. (Our notation follows that of section 4.5 of [HNT05]_.)
.. SEEALSO::
:meth:`over`
EXAMPLES:
Showing only the nodes of a binary tree, here is an example for the under operation::
sage: b1 = BinaryTree([[],[]]) sage: b2 = BinaryTree([None,[]]) sage: ascii_art((b1, b2, b1 \ b2)) ( o , o , _o_ ) ( / \ \ / \ ) ( o o o o o ) ( / \ ) ( o o )
TESTS::
sage: b1 = BinaryTree([[],[[None,[]],None]]); ascii_art([b1]) [ _o_ ] [ / \ ] [ o o ] [ / ] [ o ] [ \ ] [ o ] sage: b2 = BinaryTree([[],[None,[]]]); ascii_art([b2]) [ o ] [ / \ ] [ o o ] [ \ ] [ o ] sage: ascii_art([b1.under(b2)]) [ o_ ] [ / \ ] [ o o ] [ / \ ] [ _o_ o ] [ / \ ] [ o o ] [ / ] [ o ] [ \ ] [ o ]
The same in the labelled case::
sage: b1 = b1.canonical_labelling() sage: b2 = b2.canonical_labelling() sage: ascii_art([b1.under(b2)]) [ 2_ ] [ / \ ] [ 1 3 ] [ / \ ] [ _2_ 4 ] [ / \ ] [ 1 5 ] [ / ] [ 3 ] [ \ ] [ 4 ] """ else:
_backslash_ = under
def under_decomposition(self): r""" Return the unique maximal decomposition as an under product.
This means that the tree is cut along all edges of its leftmost path.
Beware that the factors are ordered starting from the root.
.. SEEALSO::
:meth:`comb`, :meth:`over_decomposition`
EXAMPLES::
sage: g = BinaryTree([]) sage: r = g.over(g); r [., [., .]] sage: l = g.under(g); l [[., .], .] sage: l.under_decomposition() [[., .], [., .]] sage: r.under_decomposition() == [r] True
sage: x = r.under(g).under(r).under(g) sage: ascii_art(x) o / o / \ o o / o \ o sage: x.under_decomposition() == [g,r,g,r] True """ return [] else:
def over_decomposition(self): """ Return the unique maximal decomposition as an over product.
This means that the tree is cut along all edges of its rightmost path.
Beware that the factors are ordered starting from the root.
.. SEEALSO::
:meth:`comb`, :meth:`under_decomposition`
EXAMPLES::
sage: g = BinaryTree([]) sage: r = g.over(g); r [., [., .]] sage: l = g.under(g); l [[., .], .] sage: r.over_decomposition() [[., .], [., .]] sage: l.over_decomposition() == [l] True
sage: x = g.over(l).over(l).over(g).over(g) sage: ascii_art(x) o \ _o_ / \ o o / \ o o \ o sage: x.over_decomposition() == [g,l,l,g,g] True """ return [] else:
def dendriform_shuffle(self, other): """ Return the list of terms in the dendriform product.
This is the list of all binary trees that can be obtained by identifying the rightmost path in ``self`` and the leftmost path in ``other``. Every term corresponds to a shuffle of the vertices on the rightmost path in ``self`` and the vertices on the leftmost path in ``other``.
EXAMPLES::
sage: u = BinaryTree() sage: g = BinaryTree([]) sage: l = BinaryTree([g, u]) sage: r = BinaryTree([u, g])
sage: list(g.dendriform_shuffle(g)) [[[., .], .], [., [., .]]]
sage: list(l.dendriform_shuffle(l)) [[[[[., .], .], .], .], [[[., .], [., .]], .], [[., .], [[., .], .]]]
sage: list(l.dendriform_shuffle(r)) [[[[., .], .], [., .]], [[., .], [., [., .]]]]
TESTS::
sage: list(u.dendriform_shuffle(u)) [.] sage: list(u.dendriform_shuffle(g)) [[., .]] sage: list(u.dendriform_shuffle(l)) [[[., .], .]] sage: list(u.dendriform_shuffle(r)) [[., [., .]]] sage: list(r.dendriform_shuffle(u)) [[., [., .]]] sage: list(l.dendriform_shuffle(u)) [[[., .], .]] """ else: else:
def sylvester_class(self, left_to_right=False): r""" Iterate over the sylvester class corresponding to the binary tree ``self``.
The sylvester class of a tree `T` is the set of permutations `\sigma` whose right-to-left binary search tree (a notion defined in [HNT05]_, Definition 7) is `T` after forgetting the labels. This is an equivalence class of the sylvester congruence (the congruence on words which holds two words `uacvbw` and `ucavbw` congruent whenever `a`, `b`, `c` are letters satisfying `a \leq b < c`, and extends by transitivity) on the symmetric group.
For example the following tree's sylvester class consists of the permutations `(1,3,2)` and `(3,1,2)`::
[ o ] [ / \ ] [ o o ]
(only the nodes are drawn here).
The right-to-left binary search tree of a word is constructed by an RSK-like insertion algorithm which proceeds as follows: Start with an empty labelled binary tree, and read the word from right to left. Each time a letter is read from the word, insert this letter in the existing tree using binary search tree insertion (:meth:`~sage.combinat.binary_tree.LabelledBinaryTree.binary_search_insert`). This is what the :meth:`~sage.combinat.permutation.Permutation.binary_search_tree` method computes if it is given the keyword ``left_to_right=False``.
Here are two more descriptions of the sylvester class of a binary search tree:
- The sylvester class of a binary search tree `T` is the set of all linear extensions of the poset corresponding to `T` (that is, of the poset whose Hasse diagram is `T`, with the root on top), provided that the nodes of `T` are labelled with `1, 2, \ldots, n` in a binary-search-tree way (i.e., every left descendant of a node has a label smaller than that of the node, and every right descendant of a node has a label higher than that of the node).
- The sylvester class of a binary search tree `T` (with vertex labels `1, 2, \ldots, n`) is the interval `[u, v]` in the right permutohedron order (:meth:`~sage.combinat.permutation.Permutation.permutohedron_lequal`), where `u` is the 312-avoiding permutation corresponding to `T` (:meth:`to_312_avoiding_permutation`), and where `v` is the 132-avoiding permutation corresponding to `T` (:meth:`to_132_avoiding_permutation`).
If the optional keyword variable ``left_to_right`` is set to ``True``, then the *left* sylvester class of ``self`` is returned instead. This is the set of permutations `\sigma` whose left-to-right binary search tree (that is, the result of the :meth:`~sage.combinat.permutation.Permutation.binary_search_tree` with ``left_to_right`` set to ``True``) is ``self``. It is an equivalence class of the left sylvester congruence.
.. WARNING::
This method yields the elements of the sylvester class as raw lists, not as permutations!
EXAMPLES:
Verifying the claim that the right-to-left binary search trees of the permutations in the sylvester class of a tree `t` all equal `t`::
sage: def test_bst_of_sc(n, left_to_right): ....: for t in BinaryTrees(n): ....: for p in t.sylvester_class(left_to_right=left_to_right): ....: p_per = Permutation(p) ....: tree = p_per.binary_search_tree(left_to_right=left_to_right) ....: if not BinaryTree(tree) == t: ....: return False ....: return True sage: test_bst_of_sc(4, False) True sage: test_bst_of_sc(5, False) # long time True
The same with the left-to-right version of binary search::
sage: test_bst_of_sc(4, True) True sage: test_bst_of_sc(5, True) # long time True
Checking that the sylvester class is the set of linear extensions of the poset of the tree::
sage: all( sorted(t.canonical_labelling().sylvester_class()) ....: == sorted(list(v) for v in t.canonical_labelling().to_poset().linear_extensions()) ....: for t in BinaryTrees(4) ) True
TESTS::
sage: list(BinaryTree([[],[]]).sylvester_class()) [[1, 3, 2], [3, 1, 2]] sage: bt = BinaryTree([[[],None],[[],[]]]) sage: l = list(bt.sylvester_class()); l [[1, 2, 4, 6, 5, 3], [1, 4, 2, 6, 5, 3], [1, 4, 6, 2, 5, 3], [1, 4, 6, 5, 2, 3], [4, 1, 2, 6, 5, 3], [4, 1, 6, 2, 5, 3], [4, 1, 6, 5, 2, 3], [4, 6, 1, 2, 5, 3], [4, 6, 1, 5, 2, 3], [4, 6, 5, 1, 2, 3], [1, 2, 6, 4, 5, 3], [1, 6, 2, 4, 5, 3], [1, 6, 4, 2, 5, 3], [1, 6, 4, 5, 2, 3], [6, 1, 2, 4, 5, 3], [6, 1, 4, 2, 5, 3], [6, 1, 4, 5, 2, 3], [6, 4, 1, 2, 5, 3], [6, 4, 1, 5, 2, 3], [6, 4, 5, 1, 2, 3]] sage: len(l) == Integer(bt.q_hook_length_fraction()(q=1)) True
Border cases::
sage: list(BinaryTree().sylvester_class()) [[]] sage: list(BinaryTree([]).sylvester_class()) [[1]] """ as shuffle
else:
self[1].sylvester_class(left_to_right=left_to_right)):
def is_full(self): r""" Return ``True`` if ``self`` is full, else return ``False``.
A full binary tree is a tree in which every node either has two child nodes or has two child leaves.
This is also known as *proper binary tree* or *2-tree* or *strictly binary tree*.
For example::
| __o__ | | / \ | | o o | | / \ | | o o | | / \ | | o o |
is not full but the next one is::
| ___o___ | | / \ | | __o__ o | | / \ | | o o | | / \ / \ | | o o o o |
EXAMPLES::
sage: BinaryTree([[[[],None],[None,[]]], []]).is_full() False sage: BinaryTree([[[[],[]],[[],[]]], []]).is_full() True sage: ascii_art(filter(lambda bt: bt.is_full(), BinaryTrees(5))) [ _o_ , _o_ ] [ / \ / \ ] [ o o o o ] [ / \ / \ ] [ o o o o ] """
def to_full(self): r""" Return the full binary tree constructed from ``self``.
Let `T` be a binary tree with `n` nodes. We construct a full binary tree `F` from `T` by attaching a leaf to each node of `T` which does not have 2 children. The resulting tree will have `2n + 1` nodes.
OUTPUT:
A full binary tree. See :meth:`is_full` for the definition of full.
.. SEEALSO::
:meth:`prune`
EXAMPLES::
sage: bt = BinaryTree([[None,[]],None]) sage: bt.to_full().is_full() True sage: ascii_art(bt) o / o \ o sage: ascii_art(bt.to_full()) __o___ / \ _o_ o / \ o o / \ o o
sage: bt = BinaryTree([[],[]]) sage: ascii_art(bt) o / \ o o sage: ascii_art(bt.to_full()) __o__ / \ o o / \ / \ o o o o
sage: BinaryTree(None).to_full() [., .]
"""
def prune(self): r""" Return the binary tree obtained by deleting each leaf of ``self``.
The operation of pruning is the left inverse of attaching as many leaves as possible to each node of a binary tree. That is to say, for all binary trees ``bt``, we have::
bt == bt.to_full().prune()
However, it is only a right inverse if and only if ``bt`` is a full binary tree::
bt == bt.prune().to_full()
OUTPUT:
A binary tree.
.. SEEALSO::
:meth:`to_full`
EXAMPLES::
sage: bt = BinaryTree([[[None, []], [[], []]], None]) sage: ascii_art(bt) o / __o__ / \ o o \ / \ o o o sage: ascii_art(bt.prune()) o / o / \ o o
We check the relationship with :meth:`to_full`::
sage: bt = BinaryTree([[[], [[None, []], []]], [[],[]]]) sage: bt == bt.to_full().prune() True sage: bt == bt.prune().to_full() False
sage: bt = BinaryTree([[[], []], [[], [[[], []], []]]]) sage: bt.is_full() True sage: bt == bt.prune().to_full() True
Pruning the empty tree is again the empty tree::
sage: bt = BinaryTree(None) sage: bt.prune() . """
def is_perfect(self): r""" Return ``True`` if ``self`` is perfect, else return ``False``.
A perfect binary tree is a full tree in which all leaves are at the same depth.
For example::
| ___o___ | | / \ | | __o__ o | | / \ | | o o | | / \ / \ | | o o o o |
is not perfect but the next one is::
| __o__ | | / \ | | o o | | / \ / \ | | o o o o |
EXAMPLES::
sage: lst = lambda i: filter(lambda bt: bt.is_perfect(), BinaryTrees(i)) sage: for i in range(8): ascii_art(lst(i)) # long time [ ] [ o ] [ ] [ o ] [ / \ ] [ o o ] [ ] [ ] [ ] [ __o__ ] [ / \ ] [ o o ] [ / \ / \ ] [ o o o o ] """
def is_complete(self): r""" Return ``True`` if ``self`` is complete, else return ``False``.
In a nutshell, a complete binary tree is a perfect binary tree except possibly in the last level, with all nodes in the last level "flush to the left".
In more detail: A complete binary tree (also called binary heap) is a binary tree in which every level, except possibly the last one (the deepest), is completely filled. At depth `n`, all nodes must be as far left as possible.
For example::
| ___o___ | | / \ | | __o__ o | | / \ | | o o | | / \ / \ | | o o o o |
is not complete but the following ones are::
| __o__ _o_ ___o___ | | / \ / \ / \ | | o o o o __o__ o | | / \ / \ / \ / \ / \ | | o o o o, o o , o o o o | | / \ / | | o o o |
EXAMPLES::
sage: lst = lambda i: filter(lambda bt: bt.is_complete(), BinaryTrees(i)) sage: for i in range(8): ascii_art(lst(i)) # long time [ ] [ o ] [ o ] [ / ] [ o ] [ o ] [ / \ ] [ o o ] [ o ] [ / \ ] [ o o ] [ / ] [ o ] [ _o_ ] [ / \ ] [ o o ] [ / \ ] [ o o ] [ __o__ ] [ / \ ] [ o o ] [ / \ / ] [ o o o ] [ __o__ ] [ / \ ] [ o o ] [ / \ / \ ] [ o o o o ] """ if self.is_empty(): return True # self := L ^ R dL = self[0].depth() dR = self[1].depth() # if L is perfect if self[0].is_perfect(): # if the depth of R == depth of L then R must be complete if dL == dR: return self[1].is_complete() # else R is perfect with depth equals depth of L - 1 elif dL == dR + 1: return self[1].is_perfect() return False # L is not perfect then R is perfect and the depth of L = the depth of # R + 1 return self[0].is_complete() and self[1].is_perfect() and dL == dR + 1
# Abstract class to serve as a Factory no instance are created. class BinaryTrees(UniqueRepresentation, Parent): """ Factory for binary trees.
A binary tree is a tree with at most 2 children. The binary trees considered here are also ordered (a.k.a. planar), that is to say, their children are ordered.
A full binary tree is a binary tree with no nodes with 1 child.
INPUT:
- ``size`` -- (optional) an integer - ``full`` -- (optional) a boolean
OUTPUT:
The set of all (full if ``full=True``) binary trees (of the given ``size`` if specified).
.. SEEALSO::
:class:`BinaryTree`, :meth:`BinaryTree.is_full`
EXAMPLES::
sage: BinaryTrees() Binary trees
sage: BinaryTrees(2) Binary trees of size 2
sage: BinaryTrees(full=True) Full binary trees
sage: BinaryTrees(3, full=True) Full binary trees of size 3
sage: BinaryTrees(4, full=True) Traceback (most recent call last): ... ValueError: n must be 0 or odd
.. NOTE::
This is a factory class whose constructor returns instances of subclasses.
.. NOTE::
The fact that BinaryTrees is a class instead of a simple callable is an implementation detail. It could be changed in the future and one should not rely on it. """ @staticmethod def __classcall_private__(cls, n=None, full=False): """ TESTS::
sage: from sage.combinat.binary_tree import (BinaryTrees_all, ....: BinaryTrees_size, FullBinaryTrees_all, FullBinaryTrees_size) sage: isinstance(BinaryTrees(2), BinaryTrees) True sage: isinstance(BinaryTrees(), BinaryTrees) True sage: isinstance(BinaryTrees(3, full=True), BinaryTrees) True sage: isinstance(BinaryTrees(full=True), BinaryTrees) True sage: BinaryTrees(2) is BinaryTrees_size(2) True sage: BinaryTrees(5).cardinality() 42 sage: BinaryTrees() is BinaryTrees_all() True sage: BinaryTrees(3, full=True) is FullBinaryTrees_size(3) True sage: BinaryTrees(5, full=True).cardinality() 2 sage: BinaryTrees(full=True) is FullBinaryTrees_all() True """ else: else: raise ValueError("n must be a nonnegative integer") else: else:
@cached_method def leaf(self): """ Return a leaf tree with ``self`` as parent.
EXAMPLES::
sage: BinaryTrees().leaf() .
TESTS::
sage: (BinaryTrees().leaf() is ....: sage.combinat.binary_tree.BinaryTrees_all().leaf()) True """
def from_tamari_sorting_tuple(key): """ Return a binary tree from its Tamari-sorting tuple.
See :meth:`~sage.combinat.binary_tree.BinaryTree.tamari_sorting_tuple`
INPUT:
- ``key`` -- a tuple of integers
EXEMPLES::
sage: from sage.combinat.binary_tree import from_tamari_sorting_tuple sage: t = BinaryTrees(60).random_element() sage: from_tamari_sorting_tuple(t.tamari_sorting_tuple()[0]) == t True """ # Find the root (or, rather, its index in the list of all # nodes) and call it ``i``:
from_tamari_sorting_tuple(key[i + 1:])])
################################################################# # Enumerated set of all binary trees #################################################################
class BinaryTrees_all(DisjointUnionEnumeratedSets, BinaryTrees):
def __init__(self): """ TESTS::
sage: from sage.combinat.binary_tree import BinaryTrees_all sage: B = BinaryTrees_all() sage: B.cardinality() +Infinity
sage: it = iter(B) sage: (next(it), next(it), next(it), next(it), next(it)) (., [., .], [., [., .]], [[., .], .], [., [., [., .]]]) sage: next(it).parent() Binary trees sage: B([]) [., .]
sage: B is BinaryTrees_all() True sage: TestSuite(B).run() # long time """ self, Family(NonNegativeIntegers(), BinaryTrees_size), facade=True, keepkey=False)
def _repr_(self): """ TESTS::
sage: BinaryTrees() # indirect doctest Binary trees """
def __contains__(self, x): """ TESTS::
sage: S = BinaryTrees() sage: 1 in S False sage: S([]) in S True """
def __call__(self, x=None, *args, **keywords): """ Ensure that ``None`` instead of ``0`` is passed by default.
TESTS::
sage: B = BinaryTrees() sage: B() . """
def unlabelled_trees(self): """ Return the set of unlabelled trees associated to ``self``.
EXAMPLES::
sage: BinaryTrees().unlabelled_trees() Binary trees """
def labelled_trees(self): """ Return the set of labelled trees associated to ``self``.
EXAMPLES::
sage: BinaryTrees().labelled_trees() Labelled binary trees """
def _element_constructor_(self, *args, **keywords): """ EXAMPLES::
sage: B = BinaryTrees() sage: B._element_constructor_([]) [., .] sage: B([[],[]]) # indirect doctest [[., .], [., .]] sage: B(None) # indirect doctest . """
Element = BinaryTree
################################################################# # Enumerated set of binary trees of a given size #################################################################
class BinaryTrees_size(BinaryTrees): """ The enumerated sets of binary trees of given size.
TESTS::
sage: from sage.combinat.binary_tree import BinaryTrees_size sage: for i in range(6): TestSuite(BinaryTrees_size(i)).run() """ def __init__(self, size): """ TESTS::
sage: S = BinaryTrees(3) sage: S == loads(dumps(S)) True
sage: S is BinaryTrees(3) True """ category=FiniteEnumeratedSets())
def _repr_(self): """ TESTS::
sage: BinaryTrees(3) # indirect doctest Binary trees of size 3 """
def __contains__(self, x): """ TESTS::
sage: S = BinaryTrees(3) sage: 1 in S False sage: S([[],[]]) in S True """
def _an_element_(self): """ TESTS::
sage: BinaryTrees(0).an_element() # indirect doctest . """
def cardinality(self): """ The cardinality of ``self``
This is a Catalan number.
TESTS::
sage: BinaryTrees(0).cardinality() 1 sage: BinaryTrees(5).cardinality() 42 """
def random_element(self): r""" Return a random ``BinaryTree`` with uniform probability.
This method generates a random ``DyckWord`` and then uses a bijection between Dyck words and binary trees.
EXAMPLES::
sage: BinaryTrees(5).random_element() # random [., [., [., [., [., .]]]]] sage: BinaryTrees(0).random_element() . sage: BinaryTrees(1).random_element() [., .]
TESTS::
sage: all(BinaryTrees(10).random_element() in BinaryTrees(10) for i in range(20)) True """
def __iter__(self): """ A basic generator.
.. TODO:: could be optimized.
TESTS::
sage: BinaryTrees(0).list() [.] sage: BinaryTrees(1).list() [[., .]] sage: BinaryTrees(3).list() [[., [., [., .]]], [., [[., .], .]], [[., .], [., .]], [[., [., .]], .], [[[., .], .], .]] """ else:
def _element_constructor_(self, *args, **keywords): """ EXAMPLES::
sage: S = BinaryTrees(0) sage: S([]) # indirect doctest Traceback (most recent call last): ... ValueError: wrong number of nodes sage: S(None) # indirect doctest .
sage: S = BinaryTrees(1) # indirect doctest sage: S([]) [., .] """
################################################################# # Enumerated set of all full binary trees #################################################################
class FullBinaryTrees_all(DisjointUnionEnumeratedSets, BinaryTrees): """ All full binary trees. """ def __init__(self): """ TESTS::
sage: from sage.combinat.binary_tree import FullBinaryTrees_all sage: FB = FullBinaryTrees_all() sage: FB.cardinality() +Infinity
sage: it = iter(FB) sage: (next(it), next(it), next(it), next(it), next(it)) (., [., .], [[., .], [., .]], [[., .], [[., .], [., .]]], [[[., .], [., .]], [., .]]) sage: next(it).parent() Binary trees sage: FB([]) [., .]
sage: FB is FullBinaryTrees_all() True sage: TestSuite(FB).run() # long time """ self, Family(NonNegativeIntegers(), _full_construction), facade=True, keepkey=False)
def _repr_(self): """ TESTS::
sage: BinaryTrees(full=True) Full binary trees """
def __contains__(self, x): """ TESTS::
sage: FB = BinaryTrees(full=True) sage: 1 in FB False sage: FB([]) in FB True """
def _element_constructor_(self, *args, **keywords): """ EXAMPLES::
sage: FB = BinaryTrees(full=True) sage: FB._element_constructor_([]) [., .] sage: FB([[],[]]) # indirect doctest [[., .], [., .]] sage: FB(None) # indirect doctest . sage: FB([None, []]) #indirect doctest Traceback (most recent call last): ... ValueError: not full """
################################################################# # Enumerated set of full binary trees of a given size #################################################################
def _full_construction(n): """ Helper function for the disjoint union construction.
TESTS::
sage: from sage.combinat.binary_tree import _full_construction sage: _full_construction(0) Full binary trees of size 0 sage: _full_construction(1) Full binary trees of size 1 sage: _full_construction(2) Full binary trees of size 3 sage: _full_construction(3) Full binary trees of size 5 """
class FullBinaryTrees_size(BinaryTrees): """ Full binary trees of a fixed size (number of nodes). """ def __init__(self, size): r""" TESTS::
sage: from sage.combinat.binary_tree import FullBinaryTrees_size sage: for i in range(1,6): ....: TestSuite(BinaryTrees(2*i-1, full=True)).run() """ category=FiniteEnumeratedSets())
def _repr_(self): r""" TESTS::
sage: BinaryTrees(3, full=True) Full binary trees of size 3 """
def __contains__(self, x): r""" TESTS::
sage: FB3 = BinaryTrees(3, full=True) sage: 1 in FB3 False sage: FB3([[], []]) in FB3 True sage: BinaryTree([[], []]) in FB3 True sage: BinaryTree([None, []]) in FB3 False """ and x.node_number() == self._size and x.is_full())
def _an_element_(self): r""" TESTS::
sage: BinaryTrees(0, full=True).an_element() .
sage: ascii_art(BinaryTrees(5, full=True).an_element()) _o_ / \ o o / \ o o """
def cardinality(self): r""" The cardinality of ``self``
This is a Catalan number.
TESTS::
sage: BinaryTrees(0, full=True).cardinality() 1 sage: BinaryTrees(5, full=True).cardinality() 2 sage: BinaryTrees(11, full=True).cardinality() 42 """
def random_element(self): r""" Return a random ``FullBinaryTree`` with uniform probability.
This method generates a random ``DyckWord`` of size `(s-1) / 2`, where `s` is the size of ``self``, which uses a bijection between Dyck words and binary trees to get a binary tree, and convert it to a full binary tree.
EXAMPLES::
sage: BinaryTrees(5, full=True).random_element() # random [[], [[], []]] sage: BinaryTrees(0, full=True).random_element() . sage: BinaryTrees(1, full=True).random_element() [., .]
TESTS::
sage: B = BinaryTrees(19, full=True) sage: all([B.random_element() in B for i in range(20)]) True """
def __iter__(self): """ A basic generator.
.. TODO:: could be optimized.
TESTS::
sage: BinaryTrees(0, full=True).list() [.] sage: BinaryTrees(1, full=True).list() [[., .]] sage: BinaryTrees(7, full=True).list() [[[., .], [[., .], [[., .], [., .]]]], [[., .], [[[., .], [., .]], [., .]]], [[[., .], [., .]], [[., .], [., .]]], [[[., .], [[., .], [., .]]], [., .]], [[[[., .], [., .]], [., .]], [., .]]] """ else:
def _element_constructor_(self, *args, **keywords): """ EXAMPLES::
sage: FB0 = BinaryTrees(0, full=True) sage: FB0(None) . sage: FB0([]) Traceback (most recent call last): ... ValueError: wrong number of nodes
sage: FB1 = BinaryTrees(1, full=True) sage: FB1([]) [., .]
sage: FB5 = BinaryTrees(5, full=True) sage: FB5([[], [None, [None, []]]]) Traceback (most recent call last): ... ValueError: not full """
class LabelledBinaryTree(AbstractLabelledClonableTree, BinaryTree): """ Labelled binary trees.
A labelled binary tree is a binary tree (see :class:`BinaryTree` for the meaning of this) with a label assigned to each node. The labels need not be integers, nor are they required to be distinct. ``None`` can be used as a label.
.. WARNING::
While it is possible to assign values to leaves (not just nodes) using this class, these labels are disregarded by various methods such as :meth:`~sage.combinat.abstract_tree.AbstractLabelledTree.labels`, :meth:`~sage.combinat.abstract_tree.AbstractLabelledTree.map_labels`, and (ironically) :meth:`~sage.combinat.abstract_tree.AbstractLabelledTree.leaf_labels`.
INPUT:
- ``children`` -- ``None`` (default) or a list, tuple or iterable of length `2` of labelled binary trees or convertible objects. This corresponds to the standard recursive definition of a labelled binary tree as being either a leaf, or a pair of:
- a pair of labelled binary trees, - and a label.
(The label is specified in the keyword variable ``label``; see below.)
Syntactic sugar allows leaving out all but the outermost calls of the ``LabelledBinaryTree()`` constructor, so that, e. g., ``LabelledBinaryTree([LabelledBinaryTree(None),LabelledBinaryTree(None)])`` can be shortened to ``LabelledBinaryTree([None,None])``. However, using this shorthand, it is impossible to label any vertex of the tree other than the root (because there is no way to pass a ``label`` variable without calling ``LabelledBinaryTree`` explicitly).
It is also allowed to abbreviate ``[None, None]`` by ``[]`` if one does not want to label the leaves (which one should not do anyway!).
- ``label`` -- (default: ``None``) the label to be put on the root of this tree.
- ``check`` -- (default: ``True``) whether checks should be performed or not.
.. TODO::
It is currently not possible to use ``LabelledBinaryTree()`` as a shorthand for ``LabelledBinaryTree(None)`` (in analogy to similar syntax in the ``BinaryTree`` class).
EXAMPLES::
sage: LabelledBinaryTree(None) . sage: LabelledBinaryTree(None, label="ae") # not well supported 'ae' sage: LabelledBinaryTree([]) None[., .] sage: LabelledBinaryTree([], label=3) # not well supported 3[., .] sage: LabelledBinaryTree([None, None]) None[., .] sage: LabelledBinaryTree([None, None], label=5) 5[., .] sage: LabelledBinaryTree([None, []]) None[., None[., .]] sage: LabelledBinaryTree([None, []], label=4) 4[., None[., .]] sage: LabelledBinaryTree([[], None]) None[None[., .], .] sage: LabelledBinaryTree("[[], .]", label=False) False[None[., .], .] sage: LabelledBinaryTree([None, LabelledBinaryTree([None, None], label=4)], label=3) 3[., 4[., .]] sage: LabelledBinaryTree([None, BinaryTree([None, None])], label=3) 3[., None[., .]]
sage: LabelledBinaryTree([[], None, []]) Traceback (most recent call last): ... ValueError: this is not a binary tree
sage: LBT = LabelledBinaryTree sage: t1 = LBT([[LBT([], label=2), None], None], label=4); t1 4[None[2[., .], .], .]
TESTS::
sage: t1 = LabelledBinaryTree([[None, [[],[[], None]]],[[],[]]]) sage: t2 = LabelledBinaryTree([[[],[]],[]]) sage: with t1.clone() as t1c: ....: t1c[1,1,1] = t2 sage: t1 == t1c False
We check for :trac:`16314`::
sage: t1 = LBT([ LBT([LBT([], label=2), ....: LBT([], label=5)], label=6), ....: None], label=4); t1 4[6[2[., .], 5[., .]], .] sage: class Foo(LabelledBinaryTree): ....: pass sage: t2 = Foo(t1.parent(), t1); t2 4[6[2[., .], 5[., .]], .] sage: t2.label() 4 sage: t2[0].label() 6 sage: t2.__class__, t2[0].__class__ (<class '__main__.Foo'>, <class '__main__.Foo'>) """ @staticmethod def __classcall_private__(cls, *args, **opts): """ Ensure that trees created by the sets and directly are the same and that they are instances of :class:`LabelledTree`.
TESTS::
sage: issubclass(LabelledBinaryTrees().element_class, LabelledBinaryTree) True sage: t0 = LabelledBinaryTree([[],[[], None]], label = 3) sage: t0.parent() Labelled binary trees sage: type(t0) <class 'sage.combinat.binary_tree.LabelledBinaryTrees_with_category.element_class'> """
@lazy_class_attribute def _auto_parent(cls): """ The automatic parent of the elements of this class.
When calling the constructor of an element of this class, one needs a parent. This class attribute specifies which parent is used.
EXAMPLES::
sage: LabelledBinaryTree._auto_parent Labelled binary trees sage: LabelledBinaryTree([], label = 3).parent() Labelled binary trees """
def _repr_(self): """ TESTS::
sage: LBT = LabelledBinaryTree sage: t1 = LBT([[LBT([], label=2), None], None], label=4); t1 4[None[2[., .], .], .] sage: LBT([[],[[], None]], label = 3) # indirect doctest 3[None[., .], None[None[., .], .]] """ else: else:
def _sort_key(self): """ Return a tuple encoding the labelled binary tree ``self``.
The first entry of the tuple is a pair consisting of the number of children of the root and the label of the root. Then the rest of the tuple is the concatenation of the tuples associated to these children (we view the children of a tree as trees themselves) from left to right.
This tuple characterizes the labelled tree uniquely, and can be used to sort the labelled binary trees provided that the labels belong to a type which is totally ordered.
EXAMPLES::
sage: L2 = LabelledBinaryTree([], label='a') sage: L3 = LabelledBinaryTree([], label='b') sage: T23 = LabelledBinaryTree([L2, L3], label='c') sage: T23._sort_key() ((2, 'c'), (2, 'a'), (0,), (0,), (2, 'b'), (0,), (0,)) """
def binary_search_insert(self, letter): """ Return the result of inserting a letter ``letter`` into the right strict binary search tree ``self``.
INPUT:
- ``letter`` -- any object comparable with the labels of ``self``
OUTPUT:
The right strict binary search tree ``self`` with ``letter`` inserted into it according to the binary search insertion algorithm.
.. NOTE:: ``self`` is supposed to be a binary search tree. This is not being checked!
A right strict binary search tree is defined to be a labelled binary tree such that for each node `n` with label `x`, every descendant of the left child of `n` has a label `\leq x`, and every descendant of the right child of `n` has a label `> x`. (Here, only nodes count as descendants, and every node counts as its own descendant too.) Leaves are assumed to have no labels.
Given a right strict binary search tree `t` and a letter `i`, the result of inserting `i` into `t` (denoted `Ins(i, t)` in the following) is defined recursively as follows:
- If `t` is empty, then `Ins(i, t)` is the tree with one node only, and this node is labelled with `i`.
- Otherwise, let `j` be the label of the root of `t`. If `i > j`, then `Ins(i, t)` is obtained by replacing the right child of `t` by `Ins(i, r)` in `t`, where `r` denotes the right child of `t`. If `i \leq j`, then `Ins(i, t)` is obtained by replacing the left child of `t` by `Ins(i, l)` in `t`, where `l` denotes the left child of `t`.
See, for example, [HNT05]_ for properties of this algorithm.
.. WARNING::
If `t` is nonempty, then inserting `i` into `t` does not change the root label of `t`. Hence, as opposed to algorithms like Robinson-Schensted-Knuth, binary search tree insertion involves no bumping.
EXAMPLES:
The example from Fig. 2 of [HNT05]_::
sage: LBT = LabelledBinaryTree sage: x = LBT(None) sage: x . sage: x = x.binary_search_insert("b"); x b[., .] sage: x = x.binary_search_insert("d"); x b[., d[., .]] sage: x = x.binary_search_insert("e"); x b[., d[., e[., .]]] sage: x = x.binary_search_insert("a"); x b[a[., .], d[., e[., .]]] sage: x = x.binary_search_insert("b"); x b[a[., b[., .]], d[., e[., .]]] sage: x = x.binary_search_insert("d"); x b[a[., b[., .]], d[d[., .], e[., .]]] sage: x = x.binary_search_insert("a"); x b[a[a[., .], b[., .]], d[d[., .], e[., .]]] sage: x = x.binary_search_insert("c"); x b[a[a[., .], b[., .]], d[d[c[., .], .], e[., .]]]
Other examples::
sage: LBT = LabelledBinaryTree sage: LBT(None).binary_search_insert(3) 3[., .] sage: LBT([], label = 1).binary_search_insert(3) 1[., 3[., .]] sage: LBT([], label = 3).binary_search_insert(1) 3[1[., .], .] sage: res = LBT(None) sage: for i in [3,1,5,2,4,6]: ....: res = res.binary_search_insert(i) sage: res 3[1[., 2[., .]], 5[4[., .], 6[., .]]] """ else: else:
def semistandard_insert(self, letter): """ Return the result of inserting a letter ``letter`` into the semistandard tree ``self`` using the bumping algorithm.
INPUT:
- ``letter`` -- any object comparable with the labels of ``self``
OUTPUT:
The semistandard tree ``self`` with ``letter`` inserted into it according to the bumping algorithm.
.. NOTE:: ``self`` is supposed to be a semistandard tree. This is not being checked!
A semistandard tree is defined to be a labelled binary tree such that for each node `n` with label `x`, every descendant of the left child of `n` has a label `> x`, and every descendant of the right child of `n` has a label `\geq x`. (Here, only nodes count as descendants, and every node counts as its own descendant too.) Leaves are assumed to have no labels.
Given a semistandard tree `t` and a letter `i`, the result of inserting `i` into `t` (denoted `Ins(i, t)` in the following) is defined recursively as follows:
- If `t` is empty, then `Ins(i, t)` is the tree with one node only, and this node is labelled with `i`.
- Otherwise, let `j` be the label of the root of `t`. If `i \geq j`, then `Ins(i, t)` is obtained by replacing the right child of `t` by `Ins(i, r)` in `t`, where `r` denotes the right child of `t`. If `i < j`, then `Ins(i, t)` is obtained by replacing the label at the root of `t` by `i`, and replacing the left child of `t` by `Ins(j, l)` in `t`, where `l` denotes the left child of `t`.
This algorithm is similar to the Robinson-Schensted-Knuth insertion algorithm for semistandard Young tableaux.
AUTHORS:
- Darij Grinberg (10 Nov 2013).
EXAMPLES::
sage: LBT = LabelledBinaryTree sage: x = LBT(None) sage: x . sage: x = x.semistandard_insert("b"); x b[., .] sage: x = x.semistandard_insert("d"); x b[., d[., .]] sage: x = x.semistandard_insert("e"); x b[., d[., e[., .]]] sage: x = x.semistandard_insert("a"); x a[b[., .], d[., e[., .]]] sage: x = x.semistandard_insert("b"); x a[b[., .], b[d[., .], e[., .]]] sage: x = x.semistandard_insert("d"); x a[b[., .], b[d[., .], d[e[., .], .]]] sage: x = x.semistandard_insert("a"); x a[b[., .], a[b[d[., .], .], d[e[., .], .]]] sage: x = x.semistandard_insert("c"); x a[b[., .], a[b[d[., .], .], c[d[e[., .], .], .]]]
Other examples::
sage: LBT = LabelledBinaryTree sage: LBT(None).semistandard_insert(3) 3[., .] sage: LBT([], label = 1).semistandard_insert(3) 1[., 3[., .]] sage: LBT([], label = 3).semistandard_insert(1) 1[3[., .], .] sage: res = LBT(None) sage: for i in [3,1,5,2,4,6]: ....: res = res.semistandard_insert(i) sage: res 1[3[., .], 2[5[., .], 4[., 6[., .]]]] """ else: else:
def right_rotate(self): r""" Return the result of right rotation applied to the labelled binary tree ``self``.
Right rotation on labelled binary trees is defined as follows: Let `T` be a labelled binary tree such that the left child of the root of `T` is a node. Let `C` be the right child of the root of `T`, and let `A` and `B` be the left and right children of the left child of the root of `T`. (Keep in mind that nodes of trees are identified with the subtrees consisting of their descendants.) Furthermore, let `y` be the label at the root of `T`, and `x` be the label at the left child of the root of `T`. Then, the right rotation of `T` is the labelled binary tree in which the root is labelled `x`, the left child of the root is `A`, whereas the right child of the root is a node labelled `y` whose left and right children are `B` and `C`. In pictures::
| y x | | / \ / \ | | x C -right-rotate-> A y | | / \ / \ | | A B B C |
Right rotation is the inverse operation to left rotation (:meth:`left_rotate`).
TESTS::
sage: LB = LabelledBinaryTree sage: b = LB([LB([LB([],"A"), LB([],"B")],"x"),LB([],"C")], "y"); b y[x[A[., .], B[., .]], C[., .]] sage: b.right_rotate() x[A[., .], y[B[., .], C[., .]]] """
def left_rotate(self): r""" Return the result of left rotation applied to the labelled binary tree ``self``.
Left rotation on labelled binary trees is defined as follows: Let `T` be a labelled binary tree such that the right child of the root of `T` is a node. Let `A` be the left child of the root of `T`, and let `B` and `C` be the left and right children of the right child of the root of `T`. (Keep in mind that nodes of trees are identified with the subtrees consisting of their descendants.) Furthermore, let `x` be the label at the root of `T`, and `y` be the label at the right child of the root of `T`. Then, the left rotation of `T` is the labelled binary tree in which the root is labelled `y`, the right child of the root is `C`, whereas the left child of the root is a node labelled `x` whose left and right children are `A` and `B`. In pictures::
| y x | | / \ / \ | | x C <-left-rotate- A y | | / \ / \ | | A B B C |
Left rotation is the inverse operation to right rotation (:meth:`right_rotate`).
TESTS::
sage: LB = LabelledBinaryTree sage: b = LB([LB([LB([],"A"), LB([],"B")],"x"),LB([],"C")], "y"); b y[x[A[., .], B[., .]], C[., .]] sage: b == b.right_rotate().left_rotate() True """
def heap_insert(self, l): r""" Return the result of inserting a letter ``l`` into the binary heap (tree) ``self``.
A binary heap is a labelled complete binary tree such that for each node, the label at the node is greater or equal to the label of each of its child nodes. (More precisely, this is called a max-heap.)
For example::
| _7_ | | / \ | | 5 6 | | / \ | | 3 4 |
is a binary heap.
See :wikipedia:`Binary_heap#Insert` for a description of how to insert a letter into a binary heap. The result is another binary heap.
INPUT:
- ``letter`` -- any object comparable with the labels of ``self``
.. NOTE::
``self`` is assumed to be a binary heap (tree). No check is performed.
TESTS::
sage: h = LabelledBinaryTree(None) sage: h = h.heap_insert(3); ascii_art([h]) [ 3 ] sage: h = h.heap_insert(4); ascii_art([h]) [ 4 ] [ / ] [ 3 ] sage: h = h.heap_insert(6); ascii_art([h]) [ 6 ] [ / \ ] [ 3 4 ] sage: h = h.heap_insert(2); ascii_art([h]) [ 6 ] [ / \ ] [ 3 4 ] [ / ] [ 2 ] sage: ascii_art([h.heap_insert(5)]) [ _6_ ] [ / \ ] [ 5 4 ] [ / \ ] [ 2 3 ] """
else: # if depth of L is greater than the depth of R # if L is perfect we insert in R # we insert in L # else ==> dL == dR # if R is perfect we have to insert on the leftmost leaf # ## TODO:: can be optimized... # else we insert on the right return B([L, R.heap_insert(label_insert)], label_root)
_UnLabelled = BinaryTree
class LabelledBinaryTrees(LabelledOrderedTrees): """ This is a parent stub to serve as a factory class for trees with various labels constraints. """ def _repr_(self): """ TESTS::
sage: LabelledBinaryTrees() # indirect doctest Labelled binary trees """
def _an_element_(self): """ Return a labelled binary tree.
EXAMPLES::
sage: LabelledBinaryTrees().an_element() # indirect doctest toto[42[3[., .], 3[., .]], 5[None[., .], None[., .]]] """
def unlabelled_trees(self): """ Return the set of unlabelled trees associated to ``self``.
EXAMPLES::
sage: LabelledBinaryTrees().unlabelled_trees() Binary trees
This is used to compute the shape::
sage: t = LabelledBinaryTrees().an_element().shape(); t [[[., .], [., .]], [[., .], [., .]]] sage: t.parent() Binary trees
TESTS::
sage: t = LabelledBinaryTrees().an_element() sage: t.canonical_labelling() 4[2[1[., .], 3[., .]], 6[5[., .], 7[., .]]] """
def labelled_trees(self): """ Return the set of labelled trees associated to ``self``.
EXAMPLES::
sage: LabelledBinaryTrees().labelled_trees() Labelled binary trees """
Element = LabelledBinaryTree
def binary_search_tree_shape(w, left_to_right=True): """ Direct computation of the binary search tree shape of a list of integers.
INPUT:
- ``w`` -- a list of integers
- ``left_to_right`` -- boolean (default ``True``)
OUTPUT: a non labelled binary tree
This is used under the same name as a method for permutations.
EXAMPLES::
sage: from sage.combinat.binary_tree import binary_search_tree_shape sage: binary_search_tree_shape([1,4,3,2]) [., [[[., .], .], .]] sage: binary_search_tree_shape([5,1,3,2]) [[., [[., .], .]], .]
By passing the option ``left_to_right=False`` one can have the insertion going from right to left::
sage: binary_search_tree_shape([1,6,4,2], False) [[., .], [., [., .]]]
TESTS::
sage: t = Permutations(30).random_element() sage: t.binary_search_tree().shape() == binary_search_tree_shape(t) True sage: t.binary_search_tree(False).shape() == binary_search_tree_shape(t, False) True """ else: binary_search_tree_shape(right, left_to_right)])
################################################################ # Interface attempt with species... # # Kept here for further reference when species will be improved ################################################################ # from sage.combinat.species.library import ( # CombinatorialSpecies, EmptySetSpecies, SingletonSpecies)
# BT = CombinatorialSpecies() # F = EmptySetSpecies() # N = SingletonSpecies() # BT.define(F+N*(BT*BT)) # # b3 = BT.isotypes(range(3)) # # tr = b3.list()[1]
# def BTsp_to_bintrees(bt): # """ # sage: from sage.combinat.binary_tree import BT, BTsp_to_bintrees # sage: BTsp_to_bintrees(BT.isotypes(range(5))[0]) # [., [., [., [., [., .]]]]] # sage: def spls(size): # ....: return [BTsp_to_bintrees(u) for u in BT.isotypes(range(size)).list()] # sage: spls(3) # [[., [., [., .]]], [., [[., .], .]], [[., .], [., .]], [[., [., .]], .], [[[., .], .], .]] # sage: all(spls(i) == BinaryTrees(i).list() for i in range(5)) # True # """ # if len(bt) == 0: # return BinaryTree() # elif len(bt) == 2 and len(bt[1]) == 2: # return BinaryTree(map(BTsp_to_bintrees, list(bt[1]))) # else: # raise ValueError |