# Ticket #14498: trac_14498-tree-imps-dg-rebased.patch

File trac_14498-tree-imps-dg-rebased.patch, 60.5 KB (added by chapoton, 6 years ago)
• ## sage/combinat/abstract_tree.py

# HG changeset patch
# User Frederic Chapoton <chapoton at math.univ-lyon1.fr>
# Date 1382123112 -7200
# Node ID b3162778b44eb8e6f0c04ed70e5b87ae4b009895
# Parent  20a34cc5729a9c90846e984ff1fd9b0dfe42d374
trac #14498: some improvements on trees

diff --git a/sage/combinat/abstract_tree.py b/sage/combinat/abstract_tree.py
 a incoherent with the data structure. from sage.structure.list_clone import ClonableArray from sage.rings.integer import Integer from sage.misc.misc_c import prod from sage.misc import lazy_attribute ############################################################################### # # use to load tikz in the preamble (one for *view* and one for *notebook*) from sage.misc.latex import latex class AbstractTree(object): |  / /        | | 4 5         | The algorithm explorer this tree in the following order: The algorithm explores this tree in the following order: 4,5,3,2,7,6,9,10,8,1. TESTS:: class AbstractTree(object): def tree_factorial(self): """ Returns the tree-factorial of self Return the tree-factorial of self Definition:
• ## sage/combinat/binary_tree.py

diff --git a/sage/combinat/binary_tree.py b/sage/combinat/binary_tree.py
 a """ Binary Trees This module deals with binary trees as mathematical (in particular immmutable) This module deals with binary trees as mathematical (in particular immutable) objects. .. NOTE:: AUTHORS: - Florent Hivert (2010-2011): initial implementation. References ---------- REFERENCES: .. [LodayRonco] Jean-Louis Loday and María O. Ronco. *Hopf algebra of the planar binary trees*. *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*. *The algebra of binary search trees*, :arxiv:math/0401089v2. .. [CP12] Gregory Chatel, Vivane Pons. *Counting smaller trees in the Tamari order*, :arxiv:1212.0751v1. """ #***************************************************************************** #       Copyright (C) 2010 Florent Hivert , class BinaryTree(AbstractClonableTree, C """ Binary trees. Binary trees here mean ordered (a.k.a. plane) binary trees, meaning that the children of each node are ordered. 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 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 simplified to BinaryTree([None,None]). It is also allowed to 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 simplified to BinaryTree([None,None]). It is also allowed to abbreviate [None, None] by []. - check -- (default to True) whether check for binary should be class BinaryTree(AbstractClonableTree, C 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 are the same and that they are instances of :class:BinaryTree. TESTS:: class BinaryTree(AbstractClonableTree, C @lazy_class_attribute def _auto_parent(cls): """ The automatic parent of the element of this class 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. class BinaryTree(AbstractClonableTree, C Binary trees sage: BinaryTree([None, None]).parent() Binary trees """ """ return BinaryTrees_all() def __init__(self, parent, children = None, check = True): class BinaryTree(AbstractClonableTree, C def check(self): """ Checks that self is a binary tree Check that self is a binary tree. EXAMPLES:: class BinaryTree(AbstractClonableTree, C def is_empty(self): """ Return whether self is  empty. 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:: class BinaryTree(AbstractClonableTree, C """ return not self def graph(self): def graph(self, with_leaves=True): """ Convert self to a digraph EXAMPLE:: 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. 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)] """ from sage.graphs.graph import DiGraph res = DiGraph() def rec(tr, idx): if not tr: return else: nbl = 2*tr.node_number() + 1 res.add_edges([[idx,idx+1], [idx,idx+1+nbl]]) rec(tr, idx + 1) rec(tr, idx + nbl + 1) rec(self, 0) return res if with_leaves:   # We want leaves and nodes. # Special treatment for the case when self has only 1 vertex. # In this case, rec(self, 0) would give a false result. if not self: return DiGraph([, lambda i,j: False]) res = DiGraph() # The edge set of res will be built up step by step using the # following function: def rec(tr, idx): if not tr:  # tr is a leaf. return else:  # tr is a node. nbl = 2*tr.node_number() + 1 res.add_edges([[idx,idx+1], [idx,idx+1+nbl]]) rec(tr, idx + 1) rec(tr, idx + nbl + 1) rec(self, 0) return res 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). if self.node_number() == 1: return DiGraph([, lambda i,j: False]) res = DiGraph() # The edge set of res will be built up step by step using the # following function: def rec(tr, idx): if not tr:  # tr is a leaf. return else:  # tr is a node. nbl = tr.node_number() if nbl > 0: res.add_edge([idx, idx + 1]) rec(tr, idx + 1) if tr.node_number() > 0: res.add_edge([idx, idx + nbl + 1]) rec(tr, idx + nbl + 1) rec(self, 0) return res def canonical_labelling(self,shift=1): """ 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 class BinaryTree(AbstractClonableTree, C else: return LTR(None) def show(self): def show(self, with_leaves=False): """ Show the binary tree show, with or without leaves depending on the Boolean keyword variable with_leaves. .. WARNING:: Left and right children might get interchanged in the actual picture. TESTS:: sage: t1 = BinaryTree([[], [[], None]]) sage: t1.show() """ self.graph().show(layout='tree', tree_root=0, tree_orientation="down") try: self.graph(with_leaves=with_leaves).show(layout='tree', tree_root=0, tree_orientation="down") 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 childlist Modify self so that it becomes a node with children child_list. INPUT: class BinaryTree(AbstractClonableTree, C ... ValueError: object is immutable; please change a copy instead. sage: with t.clone() as t1: ...    t1.make_node([None, None]) ....:     t1.make_node([None, None]) sage: t, t1 (., [., .]) sage: with t.clone() as t: ...    t.make_node([BinaryTree(), BinaryTree(), BinaryTree([])]) ....:     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]) ....:     t2.make_node([t1, t1]) sage: with t2.clone() as t3: ...    t3.make_node([t1, t2]) ....:     t3.make_node([t1, t2]) sage: t1, t2, t3 ([., .], [[., .], [., .]], [[., .], [[., .], [., .]]]) """ class BinaryTree(AbstractClonableTree, C def make_leaf(self): """ Modify self so that it became a leaf Modify self so that it becomes a leaf (i. e., an empty tree). .. NOTE:: self must be in a mutable state. class BinaryTree(AbstractClonableTree, C ... ValueError: object is immutable; please change a copy instead. sage: with t.clone() as t1: ...    t1.make_leaf() ....:     t1.make_leaf() sage: t, t1 ([., .], .) """ class BinaryTree(AbstractClonableTree, C 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 Tamari order on Dyck words and binary trees. The bijection is defined recursively as follows: class BinaryTree(AbstractClonableTree, C sage: BinaryTree([[[], [[], None]], [[], []]]).to_dyck_word_tamari() [1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0] """ from sage.combinat.dyck_word import DyckWord return self.to_dyck_word("L1R0") @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 Return the Dyck word associated with self using the given map. 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 l and r. Dyck words associated with the trees l and r. EXAMPLES:: class BinaryTree(AbstractClonableTree, C @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: 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 class BinaryTree(AbstractClonableTree, C 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 sylester class of the tree. This permutation is order called the sylvester class of the tree. This permutation is the minimal element of this sylvester class. EXAMPLES:: class BinaryTree(AbstractClonableTree, C return Permutation(self._postfix_word()) @combinatorial_map(name="To complete tree") def as_ordered_tree(self,with_leaves=True): 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. are transformed into actual nodes, but this can be avoided by setting the optional variable with_leaves to False. EXAMPLES:: class BinaryTree(AbstractClonableTree, C if with_leaves: children = [child.as_ordered_tree(with_leaves) for child in self] else: if not self: raise ValueError("The empty binary tree cannot be made into an ordered tree with with_leaves = False") children = [child.as_ordered_tree(with_leaves) for child in self if not child.is_empty()] if self in LabelledBinaryTrees(): from sage.combinat.ordered_tree import LabelledOrderedTree class BinaryTree(AbstractClonableTree, C return OrderedTree(children) @combinatorial_map(name="To graph") def to_undirected_graph(self, with_leaves = False): def to_undirected_graph(self, with_leaves=False): r""" Return the undirected graph obtained from the tree nodes and edges. Leafs are ignored by default but can set with_leaves to True to obtain the graph of the complete tree. 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:: class BinaryTree(AbstractClonableTree, C 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. class BinaryTree(AbstractClonableTree, C sage: BinaryTree([[],[]]).to_undirected_graph() == BinaryTree([[[],None],None]).to_undirected_graph() True """ if (not with_leaves) and (not self): # this case needs extra care :( from sage.graphs.graph import Graph return Graph([]) return self.as_ordered_tree(with_leaves).to_undirected_graph() @combinatorial_map(name="To poset") def to_poset(self, with_leaves = False, root_to_leaf=False): def to_poset(self, with_leaves=False, root_to_leaf=False): r""" Return the poset obtained by interpreting the tree as a hasse 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. Leafs are ignored by default but can set with_leaves to True to obtain the poset of the complete tree. Leaves are ignored by default, but one can set with_leaves to True to obtain the poset of the complete tree. INPUT: - with_leaves -- boolean, true if leaves should be added to the poset - root_to_leaf -- boolean, true if the poset orientation should be from root to leaves. It is false by default. - 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:: class BinaryTree(AbstractClonableTree, C 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 """ if (not with_leaves) and (not self): # this case needs extra care :( from sage.combinat.posets.posets import Poset return Poset({}) return self.as_ordered_tree(with_leaves).to_poset(root_to_leaf) @combinatorial_map(name="To 132 avoiding permutation") class BinaryTree(AbstractClonableTree, C 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 sylester class of the tree. This permutation is order called the sylvester class of the tree. This permutation is the maximal element of this sylvester class. EXAMPLES:: class BinaryTree(AbstractClonableTree, C def canopee(self): """ Return the canopee of self The *canopee* of a non empty binary tree T with n internal nodes is 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. class BinaryTree(AbstractClonableTree, C 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 [DG]_, see 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 map(lambda x:1-x, u.canopee()) == v.canopee()]) ...    for n in range(1, 5)] ....:       if map(lambda x:1-x, u.canopee()) == v.canopee()]) ....:    for n in range(1, 5)] [1, 2, 6, 22] Here is a less trivial implementation of this:: class BinaryTree(AbstractClonableTree, C 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) ....:     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] class BinaryTree(AbstractClonableTree, C REFERENCES: .. [DG] S. Dulucq and O, Guibert. Mots de piles, tableaux standards et permutations de Baxter, proceedings of Formal Power Series and Algebraic Combinatorics, 1994. .. [DG94] S. Dulucq and O. Guibert. Mots de piles, tableaux standards et permutations de Baxter, proceedings of Formal Power Series and Algebraic Combinatorics, 1994. """ if not self: raise ValueError("canopee is only defined for non empty binary trees") class BinaryTree(AbstractClonableTree, C def in_order_traversal_iter(self): """ The depth-first infix-order traversal iterator. This method iters each node following the 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:: The depth-first infix-order traversal iterator for the binary tree self. This method iters each vertex (node and leaf alike) 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____          | |    /         \         | class BinaryTree(AbstractClonableTree, C |         / \   / \      | |        d   e f   g     | the *depth-first infixe-order traversal algorithm* explores T in the following order of nodes a,1,b,2,c,3,d,4,e,5,f,6,g,7,h,8,i. ALGORITHM:: explore the left subtree manipulate the root with function node_action if the root is a node and with function leaf_action if it is a leaf explore the right subtree 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:: class BinaryTree(AbstractClonableTree, C def in_order_traversal(self, node_action=None, leaf_action=None): r""" The depth-first infix-order traversal algorithm. .. SEEALSO:: :meth:~sage.combinat.binary_tree.BinaryTree.in_order_traversal_iter() 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: class BinaryTree(AbstractClonableTree, C def tamari_greater(self): r""" The list of all trees greater than self. The list of all trees greater or equal to self in the Tamari order. This is the transitive ideal of its successors. The tree:: 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. See [CP12]_. For example, the tree:: |     __o__   | |    /     \  | class BinaryTree(AbstractClonableTree, C |  / \     /  | | o   o   o   | has these trees greater than it:: has these trees greater or equal to it:: |o          , o        , o        , o        ,  o       ,   o      ,| | \            \          \          \           \           \      | class BinaryTree(AbstractClonableTree, C def tamari_pred(self): r""" Compute the list of predecessor of self in the tamari poset. Compute the list of predecessors of self in the Tamari poset. This list is computed by all left rotate possible on its nodes. class BinaryTree(AbstractClonableTree, C def tamari_smaller(self): r""" The list of all trees smaller than self. The list of all trees smaller or equal to self in the Tamari order. This is the transitive ideal of its predecessors. The tree:: class BinaryTree(AbstractClonableTree, C |  / \     /  | | o   o   o   | has these trees smaller than it:: has these trees smaller or equal to it:: |    __o__  ,       _o_  ,        o ,         o,         o,           o | |   /     \        /   \         /           /          /            /  | class BinaryTree(AbstractClonableTree, C def tamari_succ(self): r""" Compute the list of successors of self in the tamari poset. There is the list of all trees obtains by a right rotate of Compute the list of successors of self in the Tamari poset. There is the list of all trees obtained by a right rotate of one of its nodes. The list of successor of:: The list of successors of:: |     __o__   | |    /     \  | class BinaryTree(AbstractClonableTree, C [[[[., .], .], .], .] sage: b.tamari_succ() [[[[., .], .], [., .]], [[[., .], [., .]], .], [[[., [., .]], .], .]] sage: b = B([]) sage: b.tamari_succ() [] sage: b = B([[],[]]) sage: b.tamari_succ() [[., [., [., .]]]] """ res = [] if self.is_empty(): class BinaryTree(AbstractClonableTree, C [B([g, self]) for g in self.tamari_succ()] + [B([self, d]) for d in self.tamari_succ()]) def q_hook_length_formula(self): def q_hook_length_fraction(self, q=None, q_factor=False): r""" Compute the q-hook length formula of a binary tree. If q=1 then q-hook length formula computes the number of permutations such that the shape of binary search tree associated is *self*. 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:: f_{q} (T) = \frac{[\lvert T \rvert]_q!}{\prod_{t \in T} q^{right(t)} \lvert t \rvert]_q} where \lvert T \rvert is the node number of T, t \in T the set of all subtrees of T, and right(t) the number of nodes of the right subtree of t. There are 20 permutations which give a binary tree of the 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 children 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__   | class BinaryTree(AbstractClonableTree, C |  / \     /  | | o   o   o   | by the binary search insertion algorithm. TESTS:: 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_formula()(q=1) sage: b.q_hook_length_fraction()(q=1) 20 sage: BinaryTree([[],[]]).q_hook_length_formula() (((q + 2)*q + 2)*q + 1)*q/((q + 1)*q + 1) 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 sage: test_genfun(7)  # long time True """ from sage.combinat.q_analogues import q_factorial, q_int from sage.symbolic.ring import SymbolicRing q = SymbolicRing().var('q') def product_of_subtrees(b): if b.is_empty(): return q ** 0 return (q ** (-b.node_number()) * q_int(b.node_number())) * \ product_of_subtrees(b) * product_of_subtrees(b) return q_factorial(self.node_number()) / product_of_subtrees(self) from sage.combinat.q_analogues import q_binomial if q is None: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing from sage.rings.integer_ring import ZZ basering = PolynomialRing(ZZ, 'q') q = basering.gen() else: basering = q.base_ring() if q_factor: def product_of_subtrees(b): if b.is_empty(): return basering.one() b0 = b b1 = b return q_binomial(b.node_number() - 1, b0.node_number(), q=q) * \ product_of_subtrees(b0) * product_of_subtrees(b1) * \ q ** (b1.node_number()) else: def product_of_subtrees(b): if b.is_empty(): return basering.one() b0 = b b1 = b return q_binomial(b.node_number() - 1, b0.node_number(), q=q) * \ product_of_subtrees(b0) * product_of_subtrees(b1) return product_of_subtrees(self) @combinatorial_map(name="Right rotate") def right_rotate(self): r""" Right rotation operation of tree. The right rotation is defined by;: 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 nonempty 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 nonempty 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_   | |    /                     /   \  | class BinaryTree(AbstractClonableTree, C |  /     \                               \      | | 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]) class BinaryTree(AbstractClonableTree, C [ o     o ] [      /  ] [     o   ] sage: b = BinaryTree([[[[],None],[None,[]]], []]);ascii_art([b]) sage: b = BinaryTree([[[[],None],[None,[]]], []]); ascii_art([b]) [       __o__   ] [      /     \  ] [     o       o ] class BinaryTree(AbstractClonableTree, C @combinatorial_map(name="Left rotate") def left_rotate(self): r""" Left rotation operation of tree. The left rotation is defined by;: 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 nonempty 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 nonempty 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 | |  /   \                      /  | class BinaryTree(AbstractClonableTree, C |                                 /     \  | |                                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]) class BinaryTree(AbstractClonableTree, C @combinatorial_map(name="Over operation on Binary Trees") def over(self, bt): r""" The over (/) operation defined by Loday-Ronco [LodayRonco]_. Return self over bt, where "over" is the over (/) operation defined by Loday-Ronco [LodayRonco]_. 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 node of T. See section 4.5 of [HNT05]_. .. TODO:: Fix this doc and figure out which of the different (inconsistent?) notations to follow. I don't see the "over" operation defined in [LodayRonco]_, although [HNT05]_ pretends that it is defined there. In [HNT05]_ the definition is somewhat vague about the order of the operads. Loday and Ronco seem to define "over" on page 2 of "Order structure on the algebra of permutations and of planar binary trees", but with a different choice of left/right. This is defined by:: class BinaryTree(AbstractClonableTree, C def sylvester_class(self, left_to_right=False): r""" Iterate the sylvester class corresponding to the binary tree. The sylvester class is the set of permutations corresponding to the canonical binary search tree with same shape of self. See [HNT05]_ for more. For example the following has the permutations (1,3,2) and (3,1,2) associated:: 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 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 sets 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 binary search tree of a word is constructed by an RSK-like algorithm, in which the word is read from right to left. If a left-to-right reading is to be employed instead, the left_to_right optional keyword variable should be set to True. TESTS:: sage: list(BinaryTree([[],[]]).sylvester_class()) [[1, 3, 2], [3, 1, 2]] sage: bt = BinaryTree([[[],None],[[],[]]]) class BinaryTree(AbstractClonableTree, C [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_formula()(q=1)) sage: len(l) == Integer(bt.q_hook_length_fraction()(q=1)) True Border cases:: sage: list(BinaryTree().sylvester_class()) [[]] sage: list(BinaryTree([]).sylvester_class()) [] """ if self.is_empty(): yield [] class BinaryTree(AbstractClonableTree, C def is_full(self): r""" A full binary tree is a tree in which every node other than the leaves has two children. 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*. class BinaryTree(AbstractClonableTree, C 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. class BinaryTree(AbstractClonableTree, C def is_complete(self): r""" Return True if self is complete, else return False. A complete binary tree is a perfect binary tree except possibly in the last level. class BinaryTree(AbstractClonableTree, C |  / \     / \      | | o   o   o   o     | is not complet but the following ones are:: is not complete but the following ones are:: |     __o__          _o_            ___o___     | |    /     \        /   \          /       \    | class BinaryTree(AbstractClonableTree, C EXAMPLES:: sage: lst = lambda i: filter(lambda bt: bt.is_complete(), BinaryTrees(i)) sage: for i in range(9): ascii_art(lst(i)) # long time [  ] class BinaryTrees(UniqueRepresentation, return BinaryTrees_all() else: if not (isinstance(n, (Integer, int)) and n >= 0): raise ValueError("n must be a non negative integer") raise ValueError("n must be a nonnegative integer") return BinaryTrees_size(Integer(n)) @cached_method class BinaryTrees_size(BinaryTrees): class LabelledBinaryTree(AbstractLabelledClonableTree, BinaryTree): """ The class of labelled binary tree 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.) .. TODO:: Explain how to init these. Maybe allow LabelledBinaryTree() syntax for LabelledBinaryTree(None) (in analogy to BinaryTree class). EXAMPLE:: class LabelledBinaryTree(AbstractLabelle @lazy_class_attribute def _auto_parent(cls): """ The automatic parent of the element of this class 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. class LabelledBinaryTree(AbstractLabelle def binary_search_insert(self, letter): """ Insert a letter in a binary search tree Return the result of inserting a letter letter into the right strict binary search tree self. INPUT: - letter -- any object comparable with the label 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. No check is performed. 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.) .. TODO:: Explain the algorithm. EXAMPLES:: sage: LBT = LabelledBinaryTree class LabelledBinaryTree(AbstractLabelle 3[1[., .], .] sage: res = LBT(None) sage: for i in [3,1,5,2,4,6]: ...       res = res.binary_search_insert(i) ....:     res = res.binary_search_insert(i) sage: res 3[1[., 2[., .]], 5[4[., .], 6[., .]]] """ class LabelledBinaryTree(AbstractLabelle def right_rotate(self): r""" Right rotation operation of a tree. The right rotation of a tree is defined by:: 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 nonempty 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 nonempty 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     | |    / \                    / \    | class LabelledBinaryTree(AbstractLabelle |  / \                        / \  | | A   B                      B   C | Right rotation is the inverse operation to left rotation (:meth:left_rotate). TESTS:: sage: LB = LabelledBinaryTree class LabelledBinaryTree(AbstractLabelle def left_rotate(self): r""" Left rotation operation of a tree. The left rotation of a tree is defined by:: 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 nonempty 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 nonempty 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     | |    / \                  / \    | class LabelledBinaryTree(AbstractLabelle |  / \                      / \  | | A   B                    B   C | Left rotation is the inverse operation to right rotation (:meth:right_rotate). TESTS:: sage: LB = LabelledBinaryTree class LabelledBinaryTree(AbstractLabelle def heap_insert(self, l): r""" Insert a letter in a binary heap (tree). Roughly, a binary heap will be a complete binary tree such that for each nodes the label is greater or equals to the label of its children. Return the result of inserting a letter l into the binary heap (tree) self. Roughly, a binary heap will be a labelled complete binary tree such that for each node the label is greater or equal to the label of each of its children. For example:: class LabelledBinaryTree(AbstractLabelle |  / \      | | 3   4     | is a binary heap. INPUT: - letter -- any object comparable with the label of self
• ## sage/combinat/ordered_tree.py

diff --git a/sage/combinat/ordered_tree.py b/sage/combinat/ordered_tree.py
 a from sage.combinat.combinatorial_map imp class OrderedTree(AbstractClonableTree, ClonableList): """ The class for (ordered rooted) Trees The class of (ordered rooted) trees. An ordered tree is constructed from a node called the root on which one An ordered tree is constructed from a node, called the root, on which one has grafted a possibly empty list of trees. There is a total order on the children of a node which is given by the order of the elements in the list. Note that there is no empty ordered tree. list. Note that there is no empty ordered tree (so the smallest ordered tree consists of just one node). INPUT: class OrderedTree(AbstractClonableTree, sage: tt1.__hash__() == tt2.__hash__() True Trees are usually immutable. However they inherits from :class:sage.structure.list_clone.ClonableList. So that they can be modified using the clone protocol: Trees are usually immutable. However they inherit from :class:sage.structure.list_clone.ClonableList, so that they can be modified using the clone protocol. Let us now see what this means. Trying to modify a non mutable tree raises an error:: Trying to modify a non-mutable tree raises an error:: sage: tt1 = tt2 Traceback (most recent call last): class OrderedTree(AbstractClonableTree, Here is the correct way to do it:: sage: with tt2.clone() as tt2: ...    tt2 = tt1 ....:     tt2 = tt1 sage: tt2 [[], [[], [[], []], [[], []]], [[], []]] It is also possible to append a child to a tree:: sage: with tt2.clone() as tt3: ...    tt3.append(OrderedTree([])) ....:     tt3.append(OrderedTree([])) sage: tt3 [[], [[], [[], []], [[], []]], [[], []], []] Or to insert a child in a tree:: sage: with tt2.clone() as tt3: ...    tt3.insert(2, OrderedTree([])) ....:     tt3.insert(2, OrderedTree([])) sage: tt3 [[], [[], [[], []], [[], []]], [], [[], []]] class OrderedTree(AbstractClonableTree, sage: tt1bis = OrderedTree(tt1) sage: with tt1.clone() as tt1: ...    tt1 = tt1bis ....:     tt1 = tt1bis sage: tt1 [[], [[], [[], []], [[], []]], [[], []]] sage: tt1 == tt2 class OrderedTree(AbstractClonableTree, Check that the hash value is correctly updated after modification:: sage: with tt2.clone() as tt2: ...    tt2[1,1] = tt1 ....:     tt2[1,1] = tt1 sage: tt1.__hash__() == tt2.__hash__() False """ class OrderedTree(AbstractClonableTree, def is_empty(self): """ Return if self is the empty tree Return if self is the empty tree. For ordered trees, returns always False For ordered trees, returns always False. .. NOTE:: this is different from bool(t) which returns whether t has some child or not.
• ## sage/misc/latex.py

diff --git a/sage/misc/latex.py b/sage/misc/latex.py
 a def latex_extra_preamble(): sage: from sage.misc.latex import latex_extra_preamble sage: print latex_extra_preamble() ... \usepackage{tikz} \newcommand{\ZZ}{\Bold{Z}} \newcommand{\NN}{\Bold{N}} \newcommand{\RR}{\Bold{R}}