Prufer Sequences
- class diofant.combinatorics.prufer.Prufer(*args, **kw_args)[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
>>> 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
>>> 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
See also
prufer_rank
,rank
,prev
,size
- property nodes
Returns the number of nodes in the tree.
Examples
>>> 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
>>> a = Prufer([[0, 1], [1, 2], [2, 3], [1, 4]]) >>> a.rank 36 >>> b = a.prev() >>> b Prufer((1, 2, 0)) >>> b.rank 35
See also
prufer_rank
,rank
,next
,size
- prufer_rank()[source]
Computes the rank of a Prufer sequence.
Examples
>>> a = Prufer([[0, 1], [0, 2], [0, 3]]) >>> a.prufer_rank() 0
- property prufer_repr
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 continuing until only two vertices remain. The Prufer sequence is the list of recorded nodes.
Examples
>>> 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]
See also
- property rank
Returns the rank of the Prufer sequence.
Examples
>>> p = Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]) >>> p.rank 778 >>> p.next(1).rank 779 >>> p.prev().rank 777
See also
prufer_rank
,next
,prev
,size
- property size
Return the number of possible trees of this Prufer object.
Examples
>>> Prufer([0]*4).size == Prufer([6]*4).size == 1296 True
See also
prufer_rank
,rank
,next
,prev
- 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.Examples
>>> 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]
See also
prufer_repr
returns Prufer sequence of a Prufer object.
- static to_tree(prufer)[source]
Return the tree (as a list of edges) of the given Prufer sequence.
Examples
>>> 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]]
References
See also
tree_repr
returns tree representation of a Prufer object.
- property tree_repr
Returns the tree representation of the Prufer object.
Examples
>>> 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]]
See also