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 sage.misc.sagex_ds.

AUTHORS:

  • Florent Hivert (2010-2011): initial implementation.

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](1, 2, 3, 4, 5, 6, 7) Florent Hivert, Jean-Christophe Novelli, and Jean-Yves Thibon. The algebra of binary search trees, Arxiv math/0401089v2.
[CP12](1, 2, 3) Gregory Chatel, Viviane Pons. Counting smaller trees in the Tamari order, Arxiv 1212.0751v1.
class sage.combinat.binary_tree.BinaryTree(parent, children=None, check=True)

Bases: sage.combinat.abstract_tree.AbstractClonableTree, sage.structure.list_clone.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:

  • childrenNone (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
as_ordered_tree(*args, **kwds)

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[]
canonical_labelling(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 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[., .]]]
canopee()

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 sequence 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)]
[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.
check()

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
graph(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.

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)]
in_order_traversal(node_action=None, leaf_action=None)

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 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']
in_order_traversal_iter()

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 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())
[.]
is_complete()

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(9): 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 ]
[       __o__     ]
[      /     \    ]
[     o       o   ]
[    / \     / \  ]
[   o   o   o   o ]
[  /              ]
[ o               ]
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:

sage: BinaryTree().is_empty()
True
sage: BinaryTree([]).is_empty()
False
sage: BinaryTree([[], None]).is_empty()
False
is_full()

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     ]
is_perfect()

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(10): ascii_art(lst(i)) # long time
[  ]
[ o ]
[  ]
[   o   ]
[  / \  ]
[ o   o ]
[  ]
[  ]
[  ]
[     __o__     ]
[    /     \    ]
[   o       o   ]
[  / \     / \  ]
[ o   o   o   o ]
[  ]
[  ]
left_border_symmetry(*args, **kwds)

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[., .]]]
left_right_symmetry(*args, **kwds)

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()
[., [[., .], .]]
left_rotate(*args, **kwds)

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 (right_rotate()).

See also

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
make_leaf()

Modify self so that it becomes a leaf (i. e., an empty tree).

Note

self must be in a mutable state.

See also

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
([., .], .)
make_node(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.

See also

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
([., .], [[., .], [., .]], [[., .], [[., .], [., .]]])
over(*args, **kwds)

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].)

See also

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   ]
q_hook_length_fraction(q=None, q_factor=False)

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

\[h_{q} (T) = \frac{[\lvert T \rvert]_q!}{\prod_{t \in T} [\lvert T_t \rvert]_q},\]

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 \(h_{q} (T)\):

\[h_{q} (T) = \binom{ \lvert T \rvert - 1 }{ \lvert T_1 \rvert }_q h_{q} (T_1) h_{q} (T_2),\]

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”:

\[f_{q} (T) = h_{q} (T) \cdot \prod_{t \in T} q^{\lvert T_{\mathrm{right}(t)} \rvert},\]

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
right_rotate(*args, **kwds)

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 (left_rotate()).

The right rotation operation introduced here is the one defined in Definition 2.1 of [CP12].

See also

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     ]
show(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. Moreover, for a labelled binary tree, the labels shown in the picture are not (in general) the ones given by the labelling!

Use _latex_(), view, _ascii_art_() or pretty_print for more faithful representations of the data of the tree.

TESTS:

sage: t1 = BinaryTree([[], [[], None]])
sage: t1.show()
sylvester_class(left_to_right=False)

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 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 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 left to right. Each time a letter is read from the word, insert this letter in the existing tree using binary search tree insertion (binary_search_insert()). 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],[[],[]]])
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]]
tamari_greater()

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.

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 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].

See also

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()
[[., [., [., [., .]]]], [., [., [[., .], .]]],
[., [[., .], [., .]]], [., [[., [., .]], .]],
[., [[[., .], .], .]], [[., .], [., [., .]]],
[[., .], [[., .], .]], [[., [., .]], [., .]],
[[., [., [., .]]], .], [[., [[., .], .]], .],
[[[., .], .], [., .]], [[[., .], [., .]], .],
[[[., [., .]], .], .], [[[[., .], .], .], .]]
tamari_pred()

Compute the list of predecessors of self in the Tamari poset.

This list is computed by performing all left rotates possible on its nodes.

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()
[[[., .], [., [., .]]], [., [[., .], [., .]]], [., [., [[., .], .]]]]
tamari_smaller()

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.

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 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].

See also

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()
[[[[[., .], .], .], .]]
tamari_succ()

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.

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()
[[., [., [., .]]]]
to_132_avoiding_permutation(*args, **kwds)

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
to_312_avoiding_permutation(*args, **kwds)

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
to_dyck_word(*args, **kwds)

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
to_dyck_word_tamari(*args, **kwds)

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]
to_ordered_tree_left_branch(*args, **kwds)

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()
[[], [[], []], [[], [[]]]]
to_ordered_tree_right_branch(*args, **kwds)

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()
[[[[]], [[]]], [[]], []]
to_poset(*args, **kwds)

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: bt.to_poset(with_leaves=True).cover_relations()
[[0, 2], [1, 2]]
sage: bt = BinaryTree([])
sage: bt.to_poset(with_leaves=True,root_to_leaf=True).cover_relations()
[[0, 1], [0, 2]]

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
to_undirected_graph(*args, **kwds)

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()
2[1[., .], 3[., 4[., .]]]
sage: bt.canonical_labelling().to_undirected_graph().edges()
[(1, 2, None), (2, 3, None), (3, 4, None)]
sage: bt.to_undirected_graph().edges()
[(0, 3, None), (1, 2, None), (2, 3, 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
under(*args, **kwds)

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].)

See also

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       ]
class sage.combinat.binary_tree.BinaryTrees

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

Factory for binary trees.

INPUT:

  • size – (optional) an integer

OUPUT:

  • the set of all binary trees (of the given size if specified)

EXAMPLES:

sage: BinaryTrees()
Binary trees

sage: BinaryTrees(2)
Binary trees of size 2

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.

leaf()

Return a leaf tree with self as parent.

EXAMPLES:

sage: BinaryTrees().leaf()
.

TEST:

sage: (BinaryTrees().leaf() is
....:  sage.combinat.binary_tree.BinaryTrees_all().leaf())
True
class sage.combinat.binary_tree.BinaryTrees_all

Bases: sage.sets.disjoint_union_enumerated_sets.DisjointUnionEnumeratedSets, sage.combinat.binary_tree.BinaryTrees

TESTS:

sage: from sage.combinat.binary_tree import BinaryTrees_all
sage: B = BinaryTrees_all()
sage: B.cardinality()
+Infinity

sage: it = iter(B)
sage: (it.next(), it.next(), it.next(), it.next(), it.next())
(., [., .], [., [., .]], [[., .], .], [., [., [., .]]])
sage: it.next().parent()
Binary trees
sage: B([])
[., .]

sage: B is BinaryTrees_all()
True
sage: TestSuite(B).run() # long time
Element

alias of BinaryTree

labelled_trees()

Return the set of labelled trees associated to self.

EXAMPLES:

sage: BinaryTrees().labelled_trees()
Labelled binary trees
unlabelled_trees()

Return the set of unlabelled trees associated to self.

EXAMPLES:

sage: BinaryTrees().unlabelled_trees()
Binary trees
class sage.combinat.binary_tree.BinaryTrees_size(size)

Bases: sage.combinat.binary_tree.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()
cardinality()

The cardinality of self

This is a Catalan number.

TESTS:

sage: BinaryTrees(0).cardinality()
1
sage: BinaryTrees(5).cardinality()
42
element_class()

TESTS:

sage: S = BinaryTrees(3)
sage: S.element_class
<class 'sage.combinat.binary_tree.BinaryTrees_all_with_category.element_class'>
sage: S.first().__class__ == BinaryTrees().first().__class__
True
class sage.combinat.binary_tree.LabelledBinaryTree(parent, children, label=None, check=True)

Bases: sage.combinat.abstract_tree.AbstractLabelledClonableTree, sage.combinat.binary_tree.BinaryTree

Labelled binary trees.

A labelled binary tree is a binary tree (see 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 labels(), map_labels(), and (ironically) leaf_labels().

INPUT:

  • childrenNone (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
binary_search_insert(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[., .]]]
heap_insert(l)

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 article 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     ]
left_rotate()

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 (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
right_rotate()

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 (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[., .]]]
semistandard_insert(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[., .]]]]
class sage.combinat.binary_tree.LabelledBinaryTrees(category=None)

Bases: sage.combinat.ordered_tree.LabelledOrderedTrees

This is a parent stub to serve as a factory class for trees with various labels constraints.

Element

alias of LabelledBinaryTree

labelled_trees()

Return the set of labelled trees associated to self.

EXAMPLES:

sage: LabelledBinaryTrees().labelled_trees()
Labelled binary trees
unlabelled_trees()

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[., .]]]

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