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 (20102011): 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 likeClonableArray
orClonableList
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 fromClonableArray
orClonableList
.How should this class be extended ?
A class extending
AbstractClonableTree
should satisfy the following assumptions: An instantiable class extending
AbstractClonableTree
should also extend theClonableElement
class or one of its subclasses generally, at leastClonableArray
.  To respect the Clone protocol, the
AbstractClonableTree.check()
method should be overridden by the new class.
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 extendsage.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()
 An instantiable class extending

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 fromClonableList
.
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
tolabel
.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 placeNote
self
must be in a mutable state. Seesage.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 objectOUTPUT:
None
,self
is modified in placeNote
self
must be in a mutable state. Seesage.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 callT.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()
isLabelledBinaryTrees()
andT.parent().unlabelled_trees()
isBinaryTrees()
 if
T
is an ordered labelled tree,T.parent()
isLabelledOrderedTrees()
andT.parent().unlabelled_trees()
isOrderedTrees()
 if
 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

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 noNone
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() [[], []]
 For a labelled tree

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 calliter(T)
should return an iterator on the children of the rootT
.  The
canonical_labelling
method should return the same value for trees that are considered equal (see the “canonical labellings” section in the documentation of theAbstractTree
class).  For a tree
T
the callT.parent().labelled_trees()
should return a parent for labelled trees of the same kind: for example, if
T
is a binary tree,T.parent()
isBinaryTrees()
andT.parent().labelled_trees()
isLabelledBinaryTrees()
 if
T
is an ordered tree,T.parent()
isOrderedTrees()
andT.parent().labelled_trees()
isLabelledOrderedTrees()
 if

breadth_first_order_traversal
(action=None)¶ Run the breadthfirst postorder 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 breadthfirst 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 depthfirst postorder 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 depthfirst preorder 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.)

node_number
()¶ Return the number of nodes of
self
.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
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.
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 zerobased).The root element is represented by the empty tuple
()
.See also
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
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
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 depthfirst postorder 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.)

post_order_traversal_iter
()¶ The depthfirst postorder traversal iterator.
This method iters each node following the depthfirst postorder 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 depthfirst postorder 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 depthfirst preorder 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 depthfirst preorder 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: \(1,2,3,4,5,6,7,8,9,10\).

pre_order_traversal_iter
()¶ The depthfirst preorder traversal iterator.
This method iters each node following the depthfirst preorder 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 depthfirst preorder 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 treefactorial of
self
.Definition:
The treefactorial \(T!\) 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
 For a tree

sage.combinat.abstract_tree.
from_hexacode
(ch, parent=None, label='@')¶ Transform an hexadecimal string into a tree.
INPUT:
ch
– an hexadecimal stringparent
– kind of trees to be produced. IfNone
, this will beLabelledOrderedTrees
label
– a label (default:'@'
) to be used for every vertex of the tree
See
AbstractTree.to_hexacode()
for the description of the encodingSee
_from_hexacode_aux()
for the actual codeEXAMPLES:
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()) [[[], []]]