Abstract Recursive Trees#

The purpose of this class is to help implement trees with a specific structure on the children of each node. For instance, one could want to define a tree in which each node sees its children as linearly (see the Ordered Trees module) or cyclically ordered.

Tree structures

Conceptually, one can define a tree structure from any object that can contain others. Indeed, a list can contain lists which contain lists which contain lists, and thus define a tree … The same can be done with sets, or any kind of iterable objects.

While any iterable is sufficient to encode trees, it can prove useful to have other methods available like isomorphism tests (see next section), conversions to DiGraphs objects (see as_digraph()) or computation of the number of automorphisms constrained by the structure on children. Providing such methods is the whole purpose of the AbstractTree class.

As a result, the AbstractTree class is not meant to be instantiated, but extended. It is expected that classes extending this one may also inherit from classes representing iterables, for instance ClonableArray or ClonableList

Constrained Trees

The tree built from a specific container will reflect the properties of the container. Indeed, if A is an iterable class whose elements are linearly ordered, a class B extending both of AbstractTree and A will be such that the children of a node will be linearly ordered. If A behaves like a set (i.e. if there is no order on the elements it contains), then two trees will be considered as equal if one can be obtained from the other through permutations between the children of a same node (see next section).

Paths and ID

It is expected that each element of a set of children should be identified by its index in the container. This way, any node of the tree can be identified by a word describing a path from the root node.

Canonical labellings

Equality between instances of classes extending both AbstractTree and A is entirely defined by the equality defined on the elements of A. A canonical labelling of such a tree, however, should be such that two trees a and b satisfying a == b have the same canonical labellings. On the other hand, the canonical labellings of trees a and b satisfying a != b are expected to be different.

For this reason, the values returned by the canonical_labelling method heavily depend on the data structure used for a node’s children and should be overridden by most of the classes extending AbstractTree if it is incoherent with the data structure.

Authors

  • Florent Hivert (2010-2011): initial revision

  • Frédéric Chapoton (2011): contributed some methods

class sage.combinat.abstract_tree.AbstractClonableTree#

Bases: sage.combinat.abstract_tree.AbstractTree

Abstract Clonable Tree.

An abstract class for trees with clone protocol (see list_clone). It is expected that classes extending this one may also inherit from classes like ClonableArray or ClonableList depending whether one wants to build trees where adding a child is allowed.

Note

Due to the limitation of Cython inheritance, one cannot inherit here from ClonableElement, because it would prevent us from later inheriting from ClonableArray or ClonableList.

How should this class be extended ?

A class extending AbstractClonableTree should satisfy the following assumptions:

See also the assumptions in AbstractTree.

check()#

Check that self is a correct tree.

This method does nothing. It is implemented here because many extensions of AbstractClonableTree also extend sage.structure.list_clone.ClonableElement, which requires it.

It should be overridden in subclasses in order to check that the characterizing property of the respective kind of tree holds (eg: two children for binary trees).

EXAMPLES:

sage: OrderedTree([[],[[]]]).check()
sage: BinaryTree([[],[[],[]]]).check()
class sage.combinat.abstract_tree.AbstractLabelledClonableTree(parent, children, label=None, check=True)#

Bases: sage.combinat.abstract_tree.AbstractLabelledTree, sage.combinat.abstract_tree.AbstractClonableTree

Abstract Labelled Clonable Tree

This class takes care of modification for the label by the clone protocol.

Note

Due to the limitation of Cython inheritance, one cannot inherit here from ClonableArray, because it would prevent us to inherit later from ClonableList.

map_labels(f)#

Apply the function \(f\) to the labels of self

This method returns a copy of self on which the function \(f\) has been applied on all labels (a label \(x\) is replaced by \(f(x)\)).

EXAMPLES:

sage: LT = LabelledOrderedTree
sage: t = LT([LT([],label=1),LT([],label=7)],label=3); t
3[1[], 7[]]
sage: t.map_labels(lambda z:z+1)
4[2[], 8[]]

sage: LBT = LabelledBinaryTree
sage: bt = LBT([LBT([],label=1),LBT([],label=4)],label=2); bt
2[1[., .], 4[., .]]
sage: bt.map_labels(lambda z:z+1)
3[2[., .], 5[., .]]
set_label(path, label)#

Change the label of subtree indexed by path to label.

INPUT:

  • path – None (default) or a path (list or tuple of children

    index in the tree)

  • label – any sage object

OUTPUT: Nothing, self is modified in place

Note

self must be in a mutable state. See sage.structure.list_clone for more details about mutability.

EXAMPLES:

sage: t = LabelledOrderedTree([[],[[],[]]])
sage: t.set_label((0,), 4)
Traceback (most recent call last):
...
ValueError: object is immutable; please change a copy instead.
sage: with t.clone() as t:
....:     t.set_label((0,), 4)
sage: t
None[4[], None[None[], None[]]]
sage: with t.clone() as t:
....:     t.set_label((1,0), label = 42)
sage: t
None[4[], None[42[], None[]]]

Todo

Do we want to implement the following syntactic sugar:

with t.clone() as tt:
    tt.labels[1,2] = 3 ?
set_root_label(label)#

Set the label of the root of self.

INPUT: label – any Sage object

OUTPUT: None, self is modified in place

Note

self must be in a mutable state. See sage.structure.list_clone for more details about mutability.

EXAMPLES:

sage: t = LabelledOrderedTree([[],[[],[]]])
sage: t.set_root_label(3)
Traceback (most recent call last):
...
ValueError: object is immutable; please change a copy instead.
sage: with t.clone() as t:
....:     t.set_root_label(3)
sage: t.label()
3
sage: t
3[None[], None[None[], None[]]]

This also works for binary trees:

sage: bt = LabelledBinaryTree([[],[]])
sage: bt.set_root_label(3)
Traceback (most recent call last):
...
ValueError: object is immutable; please change a copy instead.
sage: with bt.clone() as bt:
....:     bt.set_root_label(3)
sage: bt.label()
3
sage: bt
3[None[., .], None[., .]]
class sage.combinat.abstract_tree.AbstractLabelledTree(parent, children, label=None, check=True)#

Bases: sage.combinat.abstract_tree.AbstractTree

Abstract Labelled Tree.

Typically a class for labelled trees is constructed by inheriting from a class for unlabelled trees and AbstractLabelledTree.

How should this class be extended ?

A class extending AbstractLabelledTree should respect the following assumptions:

  • For a labelled tree T the call T.parent().unlabelled_trees() should return a parent for unlabelled trees of the same kind: for example,

    • if T is a binary labelled tree, T.parent() is LabelledBinaryTrees() and T.parent().unlabelled_trees() is BinaryTrees()

    • if T is an ordered labelled tree, T.parent() is LabelledOrderedTrees() and T.parent().unlabelled_trees() is OrderedTrees()

  • In the same vein, the class of T should contain an attribute _UnLabelled which should be the class for the corresponding unlabelled trees.

See also the assumptions in AbstractTree.

See also

AbstractTree

as_digraph()#

Return a directed graph version of self.

Warning

At this time, the output makes sense only if self is a labelled binary tree with no repeated labels and no None labels.

EXAMPLES:

sage: LT = LabelledOrderedTrees()
sage: t1 = LT([LT([],label=6),LT([],label=1)],label=9)
sage: t1.as_digraph()
Digraph on 3 vertices

sage: t = BinaryTree([[None, None],[[],None]])
sage: lt = t.canonical_labelling()
sage: lt.as_digraph()
Digraph on 4 vertices
label(path=None)#

Return the label of self.

INPUT:

  • path – None (default) or a path (list or tuple of children index in the tree)

OUTPUT: the label of the subtree indexed by path

EXAMPLES:

sage: t = LabelledOrderedTree([[],[]], label = 3)
sage: t.label()
3
sage: t[0].label()
sage: t = LabelledOrderedTree([LabelledOrderedTree([], 5),[]], label = 3)
sage: t.label()
3
sage: t[0].label()
5
sage: t[1].label()
sage: t.label([0])
5
labels()#

Return the list of labels of self.

EXAMPLES:

sage: LT = LabelledOrderedTree
sage: t = LT([LT([],label='b'),LT([],label='c')],label='a')
sage: t.labels()
['a', 'b', 'c']

sage: LBT = LabelledBinaryTree
sage: LBT([LBT([],label=1),LBT([],label=4)],label=2).labels()
[2, 1, 4]
leaf_labels()#

Return the list of labels of the leaves of self.

In case of a labelled binary tree, these “leaves” are not actually the leaves of the binary trees, but the nodes whose both children are leaves!

EXAMPLES:

sage: LT = LabelledOrderedTree
sage: t = LT([LT([],label='b'),LT([],label='c')],label='a')
sage: t.leaf_labels()
['b', 'c']

sage: LBT = LabelledBinaryTree
sage: bt = LBT([LBT([],label='b'),LBT([],label='c')],label='a')
sage: bt.leaf_labels()
['b', 'c']
sage: LBT([], label='1').leaf_labels()
['1']
sage: LBT(None).leaf_labels()
[]
shape()#

Return the unlabelled tree associated to self.

EXAMPLES:

sage: t = LabelledOrderedTree([[],[[]]], label = 25).shape(); t
[[], [[]]]

sage: LabelledBinaryTree([[],[[],[]]], label = 25).shape()
[[., .], [[., .], [., .]]]

sage: LRT = LabelledRootedTree
sage: tb = LRT([],label='b')
sage: LRT([tb, tb], label='a').shape()
[[], []]
class sage.combinat.abstract_tree.AbstractTree#

Bases: object

Abstract Tree.

There is no data structure defined here, as this class is meant to be extended, not instantiated.

How should this class be extended?

A class extending AbstractTree should respect several assumptions:

  • For a tree T, the call iter(T) should return an iterator on the children of the root T.

  • The canonical_labelling method should return the same value for trees that are considered equal (see the “canonical labellings” section in the documentation of the AbstractTree class).

  • For a tree T the call T.parent().labelled_trees() should return a parent for labelled trees of the same kind: for example,

    • if T is a binary tree, T.parent() is BinaryTrees() and T.parent().labelled_trees() is LabelledBinaryTrees()

    • if T is an ordered tree, T.parent() is OrderedTrees() and T.parent().labelled_trees() is LabelledOrderedTrees()

breadth_first_order_traversal(action=None)#

Run the breadth-first post-order traversal algorithm and subject every node encountered to some procedure action. The algorithm is:

queue <- [ root ];
while the queue is not empty:
    node <- pop( queue );
    manipulate the node;
    prepend to the queue the list of all subtrees of
        the node (from the rightmost to the leftmost).

INPUT:

  • action – (optional) a function which takes a node as input, and does something during the exploration

OUTPUT:

None. (This is not an iterator.)

EXAMPLES:

For example, on the following binary tree \(b\):

|   ___3____      |
|  /        \     |
| 1         _7_   |
|  \       /   \  |
|   2     5     8 |
|        / \      |
|       4   6     |

the breadth-first order traversal algorithm explores \(b\) in the following order of nodes: \(3,1,7,2,5,8,4,6\).

canonical_labelling(shift=1)#

Return a labelled version of self.

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: t = OrderedTree([[], [[], [[], []], [[], []]], [[], []]])
sage: t.canonical_labelling()
1[2[], 3[4[], 5[6[], 7[]], 8[9[], 10[]]], 11[12[], 13[]]]

sage: BinaryTree([]).canonical_labelling()
1[., .]
sage: BinaryTree([[],[[],[]]]).canonical_labelling()
2[1[., .], 4[3[., .], 5[., .]]]
depth()#

Return the depth of self.

EXAMPLES:

sage: OrderedTree().depth()
1
sage: OrderedTree([]).depth()
1
sage: OrderedTree([[],[]]).depth()
2
sage: OrderedTree([[],[[]]]).depth()
3
sage: OrderedTree([[], [[], [[], []], [[], []]], [[], []]]).depth()
4

sage: BinaryTree().depth()
0
sage: BinaryTree([[],[[],[]]]).depth()
3
iterative_post_order_traversal(action=None)#

Run the depth-first post-order traversal algorithm (iterative implementation) and subject every node encountered to some procedure action. The algorithm is:

explore each subtree (by the algorithm) from the
    leftmost one to the rightmost one;
then manipulate the root with function `action` (in the
    case of a binary tree, only if the root is not a leaf).

INPUT:

  • action – (optional) a function which takes a node as input, and does something during the exploration

OUTPUT:

None. (This is not an iterator.)

See also

iterative_pre_order_traversal(action=None)#

Run the depth-first pre-order traversal algorithm (iterative implementation) and subject every node encountered to some procedure action. The algorithm is:

manipulate the root with function `action` (in the case
    of a binary tree, only if the root is not a leaf);
then explore each subtree (by the algorithm) from the
    leftmost one to the rightmost one.

INPUT:

  • action – (optional) a function which takes a node as input, and does something during the exploration

OUTPUT:

None. (This is not an iterator.)

See also

node_number()#

Return the number of nodes of self.

See also

node_number_at_depth(), node_number_to_the_right()

EXAMPLES:

sage: OrderedTree().node_number()
1
sage: OrderedTree([]).node_number()
1
sage: OrderedTree([[],[]]).node_number()
3
sage: OrderedTree([[],[[]]]).node_number()
4
sage: OrderedTree([[], [[], [[], []], [[], []]], [[], []]]).node_number()
13

EXAMPLES:

sage: BinaryTree(None).node_number()
0
sage: BinaryTree([]).node_number()
1
sage: BinaryTree([[], None]).node_number()
2
sage: BinaryTree([[None, [[], []]], None]).node_number()
5
node_number_at_depth(depth)#

Return the number of nodes at a given depth.

This counts all nodes that are at the given depth.

Here the root is considered to have depth 0.

INPUT:

  • depth – an integer

See also

node_number(), node_number_to_the_right(), paths_at_depth()

EXAMPLES:

sage: T = OrderedTree([[[], [[]]], [[], [[[]]]], []])
sage: ascii_art(T)
    ___o____
   /    /  /
  o_   o_ o
 / /  / /
o o  o o
  |    |
  o    o
       |
       o
sage: [T.node_number_at_depth(i) for i in range(6)]
[1, 3, 4, 2, 1, 0]
node_number_to_the_right(path)#

Return the number of nodes at the same depth and to the right of the node identified by path.

This counts the nodes that are at the same depth as the given one, and strictly to its right.

See also

node_number(), node_number_at_depth(), paths_to_the_right()

EXAMPLES:

sage: T = OrderedTree([[[], [[]]], [[], [[[]]]], []])
sage: ascii_art(T)
    ___o____
   /    /  /
  o_   o_ o
 / /  / /
o o  o o
  |    |
  o    o
       |
       o
sage: T.node_number_to_the_right(())
0
sage: T.node_number_to_the_right((0,))
2
sage: T.node_number_to_the_right((0,1))
2
sage: T.node_number_to_the_right((0,1,0))
1

sage: T = OrderedTree([])
sage: T.node_number_to_the_right(())
0
paths()#

Return a generator for all paths to nodes of self.

OUTPUT:

This method returns a list of sequences of integers. Each of these sequences represents a path from the root node to some node. For instance, \((1, 3, 2, 5, 0, 3)\) represents the node obtained by choosing the 1st child of the root node (in the ordering returned by iter), then the 3rd child of its child, then the 2nd child of the latter, etc. (where the labelling of the children is zero-based).

The root element is represented by the empty tuple ().

See also

paths_at_depth(), paths_to_the_right()

EXAMPLES:

sage: list(OrderedTree([]).paths())
[()]
sage: list(OrderedTree([[],[[]]]).paths())
[(), (0,), (1,), (1, 0)]

sage: list(BinaryTree([[],[[],[]]]).paths())
[(), (0,), (1,), (1, 0), (1, 1)]
paths_at_depth(depth, path=[])#

Return a generator for all paths at a fixed depth.

This iterates over all paths for nodes that are at the given depth.

Here the root is considered to have depth 0.

INPUT:

  • depth – an integer

  • path – optional given path (as a list) used in the recursion

Warning

The path option should not be used directly.

See also

paths(), paths_to_the_right(), node_number_at_depth()

EXAMPLES:

sage: T = OrderedTree([[[], [[], [[]]]], [], [[[],[]]], [], []])
sage: ascii_art(T)
     ______o_______
    /    /   /  / /
  _o__  o   o  o o
 /   /      |
o   o_      o_
   / /     / /
  o o     o o
    |
    o
sage: list(T.paths_at_depth(0))
[()]
sage: list(T.paths_at_depth(2))
[(0, 0), (0, 1), (2, 0)]
sage: list(T.paths_at_depth(4))
[(0, 1, 1, 0)]
sage: list(T.paths_at_depth(5))
[]

sage: T2 = OrderedTree([])
sage: list(T2.paths_at_depth(0))
[()]
paths_to_the_right(path)#

Return a generator of paths for all nodes at the same depth and to the right of the node identified by path.

This iterates over the paths for nodes that are at the same depth as the given one, and strictly to its right.

INPUT:

  • path – any path in the tree

See also

paths(), paths_at_depth(), node_number_to_the_right()

EXAMPLES:

sage: T = OrderedTree([[[], [[]]], [[], [[[]]]], []])
sage: ascii_art(T)
    ___o____
   /    /  /
  o_   o_ o
 / /  / /
o o  o o
  |    |
  o    o
       |
       o
sage: g = T.paths_to_the_right(())
sage: list(g)
[]

sage: g = T.paths_to_the_right((0,))
sage: list(g)
[(1,), (2,)]

sage: g = T.paths_to_the_right((0,1))
sage: list(g)
[(1, 0), (1, 1)]

sage: g = T.paths_to_the_right((0,1,0))
sage: list(g)
[(1, 1, 0)]

sage: g = T.paths_to_the_right((1,2))
sage: list(g)
[]
post_order_traversal(action=None)#

Run the depth-first post-order traversal algorithm (recursive implementation) and subject every node encountered to some procedure action. The algorithm is:

explore each subtree (by the algorithm) from the
    leftmost one to the rightmost one;
then manipulate the root with function `action` (in the
    case of a binary tree, only if the root is not a leaf).

INPUT:

  • action – (optional) a function which takes a node as input, and does something during the exploration

OUTPUT:

None. (This is not an iterator.)

See also

post_order_traversal_iter()#

The depth-first post-order traversal iterator.

This method iters each node following the depth-first post-order traversal algorithm (recursive implementation). The algorithm is:

explore each subtree (by the algorithm) from the
    leftmost one to the rightmost one;
then yield the root (in the case of binary trees, only if
    it is not a leaf).

EXAMPLES:

For example on the following binary tree \(b\):

|   ___3____      |
|  /        \     |
| 1         _7_   |
|  \       /   \  |
|   2     5     8 |
|        / \      |
|       4   6     |

(only the nodes are shown), the depth-first post-order traversal algorithm explores \(b\) in the following order of nodes: \(2,1,4,6,5,8,7,3\).

For another example, consider the labelled tree:

|     __1____ |
|    /  /   / |
|   2  6   8_ |
|   |  |  / / |
|   3_ 7 9 10 |
|  / /        |
| 4 5         |

The algorithm explores this tree in the following order: \(4,5,3,2,7,6,9,10,8,1\).

pre_order_traversal(action=None)#

Run the depth-first pre-order traversal algorithm (recursive implementation) and subject every node encountered to some procedure action. The algorithm is:

manipulate the root with function `action` (in the case
    of a binary tree, only if the root is not a leaf);
then explore each subtree (by the algorithm) from the
    leftmost one to the rightmost one.

INPUT:

  • action – (optional) a function which takes a node as input, and does something during the exploration

OUTPUT:

None. (This is not an iterator.)

EXAMPLES:

For example, on the following binary tree \(b\):

|   ___3____      |
|  /        \     |
| 1         _7_   |
|  \       /   \  |
|   2     5     8 |
|        / \      |
|       4   6     |

the depth-first pre-order traversal algorithm explores \(b\) in the following order of nodes: \(3,1,2,7,5,4,6,8\).

Another example:

|     __1____ |
|    /  /   / |
|   2  6   8_ |
|   |  |  / / |
|   3_ 7 9 10 |
|  / /        |
| 4 5         |

The algorithm explores this tree in the following order: \(1,2,3,4,5,6,7,8,9,10\).

See also

pre_order_traversal_iter()#

The depth-first pre-order traversal iterator.

This method iters each node following the depth-first pre-order traversal algorithm (recursive implementation). The algorithm is:

yield the root (in the case of binary trees, if it is not
    a leaf);
then explore each subtree (by the algorithm) from the
    leftmost one to the rightmost one.

EXAMPLES:

For example, on the following binary tree \(b\):

|   ___3____      |
|  /        \     |
| 1         _7_   |
|  \       /   \  |
|   2     5     8 |
|        / \      |
|       4   6     |

(only the nodes shown), the depth-first pre-order traversal algorithm explores \(b\) in the following order of nodes: \(3,1,2,7,5,4,6,8\).

Another example:

|     __1____ |
|    /  /   / |
|   2  6   8_ |
|   |  |  / / |
|   3_ 7 9 10 |
|  / /        |
| 4 5         |

The algorithm explores this labelled tree in the following order: \(1,2,3,4,5,6,7,8,9,10\).

subtrees()#

Return a generator for all nonempty subtrees of self.

The number of nonempty subtrees of a tree is its number of nodes. (The word “nonempty” makes a difference only in the case of binary trees. For ordered trees, for example, all trees are nonempty.)

EXAMPLES:

sage: list(OrderedTree([]).subtrees())
[[]]
sage: list(OrderedTree([[],[[]]]).subtrees())
[[[], [[]]], [], [[]], []]

sage: list(OrderedTree([[],[[]]]).canonical_labelling().subtrees())
[1[2[], 3[4[]]], 2[], 3[4[]], 4[]]

sage: list(BinaryTree([[],[[],[]]]).subtrees())
[[[., .], [[., .], [., .]]], [., .], [[., .], [., .]], [., .], [., .]]

sage: v = BinaryTree([[],[]])
sage: list(v.canonical_labelling().subtrees())
[2[1[., .], 3[., .]], 1[., .], 3[., .]]
to_hexacode()#

Transform a tree into an hexadecimal string.

The definition of the hexacode is recursive. The first letter is the valence of the root as an hexadecimal (up to 15), followed by the concatenation of the hexacodes of the subtrees.

This method only works for trees where every vertex has valency at most 15.

See from_hexacode() for the reverse transformation.

EXAMPLES:

sage: from sage.combinat.abstract_tree import from_hexacode
sage: LT = LabelledOrderedTrees()
sage: from_hexacode('2010', LT).to_hexacode()
'2010'
sage: LT.an_element().to_hexacode()
'3020010'
sage: t = from_hexacode('a0000000000000000', LT)
sage: t.to_hexacode()
'a0000000000'

sage: OrderedTrees(6).an_element().to_hexacode()
'500000'
tree_factorial()#

Return the tree-factorial of self.

Definition:

The tree-factorial \(T!\) of a tree \(T\) is the product \(\prod_{v\in T}\#\mbox{children}(v)\).

EXAMPLES:

sage: LT = LabelledOrderedTrees()
sage: t = LT([LT([],label=6),LT([],label=1)],label=9)
sage: t.tree_factorial()
3

sage: BinaryTree([[],[[],[]]]).tree_factorial()
15
sage.combinat.abstract_tree.from_hexacode(ch, parent=None, label='@')#

Transform an hexadecimal string into a tree.

INPUT:

  • ch – an hexadecimal string

  • parent – kind of trees to be produced. If None, this will be LabelledOrderedTrees

  • label – a label (default: '@') to be used for every vertex of the tree

See AbstractTree.to_hexacode() for the description of the encoding

See _from_hexacode_aux() for the actual code

EXAMPLES:

sage: from sage.combinat.abstract_tree import from_hexacode
sage: from_hexacode('12000', LabelledOrderedTrees())
@[@[@[], @[]]]
sage: from_hexacode('12000')
@[@[@[], @[]]]

sage: from_hexacode('1200', LabelledOrderedTrees())
@[@[@[], @[]]]

It can happen that only a prefix of the word is used:

sage: from_hexacode('a'+14*'0', LabelledOrderedTrees())
@[@[], @[], @[], @[], @[], @[], @[], @[], @[], @[]]

One can choose the label:

sage: from_hexacode('1200', LabelledOrderedTrees(), label='o')
o[o[o[], o[]]]

One can also create other kinds of trees:

sage: from_hexacode('1200', OrderedTrees())
[[[], []]]