# Prufer Sequences¶

class sympy.combinatorics.prufer.Prufer[source]

The Prufer correspondence is an algorithm that describes the bijection between labeled trees and the Prufer code. A Prufer code of a labeled tree is unique up to isomorphism and has a length of n - 2.

Prufer sequences were first used by Heinz Prufer to give a proof of Cayley’s formula.

References

static edges(*runs)[source]

Return a list of edges and the number of nodes from the given runs that connect nodes in an integer-labelled tree.

All node numbers will be shifted so that the minimum node is 0. It is not a problem if edges are repeated in the runs; only unique edges are returned. There is no assumption made about what the range of the node labels should be, but all nodes from the smallest through the largest must be present.

Examples

>>> from sympy.combinatorics.prufer import Prufer
>>> Prufer.edges([1, 2, 3], [2, 4, 5]) # a T
([[0, 1], [1, 2], [1, 3], [3, 4]], 5)


Duplicate edges are removed:

>>> Prufer.edges([0, 1, 2, 3], [1, 4, 5], [1, 4, 6]) # a K
([[0, 1], [1, 2], [1, 4], [2, 3], [4, 5], [4, 6]], 7)

next(delta=1)[source]

Generates the Prufer sequence that is delta beyond the current one.

Examples

>>> from sympy.combinatorics.prufer import Prufer
>>> a = Prufer([[0, 1], [0, 2], [0, 3]])
>>> b = a.next(1) # == a.next()
>>> b.tree_repr
[[0, 2], [0, 1], [1, 3]]
>>> b.rank
1

nodes[source]

Returns the number of nodes in the tree.

Examples

>>> from sympy.combinatorics.prufer import Prufer
>>> Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).nodes
6
>>> Prufer([1, 0, 0]).nodes
5

prev(delta=1)[source]

Generates the Prufer sequence that is -delta before the current one.

Examples

>>> from sympy.combinatorics.prufer import Prufer
>>> a = Prufer([[0, 1], [1, 2], [2, 3], [1, 4]])
>>> a.rank
36
>>> b = a.prev()
>>> b
Prufer([1, 2, 0])
>>> b.rank
35

prufer_rank()[source]

Computes the rank of a Prufer sequence.

Examples

>>> from sympy.combinatorics.prufer import Prufer
>>> a = Prufer([[0, 1], [0, 2], [0, 3]])
>>> a.prufer_rank()
0

prufer_repr[source]

Returns Prufer sequence for the Prufer object.

This sequence is found by removing the highest numbered vertex, recording the node it was attached to, and continuuing until only two verices remain. The Prufer sequence is the list of recorded nodes.

to_prufer

Examples

>>> from sympy.combinatorics.prufer import Prufer
>>> Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).prufer_repr
[3, 3, 3, 4]
>>> Prufer([1, 0, 0]).prufer_repr
[1, 0, 0]

rank[source]

Returns the rank of the Prufer sequence.

Examples

>>> from sympy.combinatorics.prufer import Prufer
>>> p = Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]])
>>> p.rank
778
>>> p.next(1).rank
779
>>> p.prev().rank
777

size[source]

Return the number of possible trees of this Prufer object.

Examples

>>> from sympy.combinatorics.prufer import Prufer
>>> Prufer([0]*4).size == Prufer([6]*4).size == 1296
True

static to_prufer(tree, n)[source]

Return the Prufer sequence for a tree given as a list of edges where n is the number of nodes in the tree.

prufer_repr
returns Prufer sequence of a Prufer object.

Examples

>>> from sympy.combinatorics.prufer import Prufer
>>> a = Prufer([[0, 1], [0, 2], [0, 3]])
>>> a.prufer_repr
[0, 0]
>>> Prufer.to_prufer([[0, 1], [0, 2], [0, 3]], 4)
[0, 0]

static to_tree(prufer)[source]

Return the tree (as a list of edges) of the given Prufer sequence.

tree_repr
returns tree representation of a Prufer object.

References

Examples

>>> from sympy.combinatorics.prufer import Prufer
>>> a = Prufer([0, 2], 4)
>>> a.tree_repr
[[0, 1], [0, 2], [2, 3]]
>>> Prufer.to_tree([0, 2])
[[0, 1], [0, 2], [2, 3]]

tree_repr[source]

Returns the tree representation of the Prufer object.

to_tree

Examples

>>> from sympy.combinatorics.prufer import Prufer
>>> Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).tree_repr
[[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]
>>> Prufer([1, 0, 0]).tree_repr
[[1, 2], [0, 1], [0, 3], [0, 4]]

classmethod unrank(rank, n)[source]

Finds the unranked Prufer sequence.

Examples

>>> from sympy.combinatorics.prufer import Prufer
>>> Prufer.unrank(0, 4)
Prufer([0, 0])


Polyhedron

Subsets