In combinatorial mathematics, the Prüfer sequence (also Prüfer code or Prüfer numbers) of a labeled tree is a unique sequence associated with the tree. The sequence for a tree on vertices has length , and can be generated by a simple iterative algorithm. Prüfer sequences were first used by Heinz Prüfer to prove Cayley's formula in 1918.
Algorithm to convert a tree into a Prüfer sequence One can generate a labeled tree's Prüfer sequence by iteratively removing vertices from the tree until only two vertices remain. Specifically, consider a labeled tree with vertices . At step , remove the leaf with the smallest label and set the -th element of the Prüfer sequence to be the label of this leaf's neighbour.
The Prüfer sequence of a labeled tree is unique and has length .
Both coding and decoding can be reduced to integer radix sorting and parallelized.
Example
Consider the above algorithm run on the tree shown to the right. Initially, vertex 1 is the leaf with the smallest label, so it is removed first and 4 is put in the Prüfer sequence. Vertices 2 and 3 are removed next, so 4 is added twice more. Vertex 4 is now a leaf and has the smallest label, so it is removed and we append 5 to the sequence. We are left with only two vertices, so we stop. The tree's sequence is [4,4,4,5].
Algorithm to convert a Prüfer sequence into a tree
Let [a, a, ..., a[n]] be a Prüfer sequence:
The tree will have n+2 nodes, numbered from 1 to n+2. For each node set its degree to the number of times it appears in the sequence plus 1. For instance, in pseudo-code:
Convert-Prüfer-to-Tree() 1 ← length[] 2 ← a graph with + 2 isolated nodes, numbered 1 to + 2 3 degree ← an array of integers 4 for each node in do 5 degree[] ← 1 6 for each value in do 7 degree[] ← degree[] + 1
Next, for each number in the sequence a[i], find the first (lowest-numbered) node, j, with degree equal to 1, add the edge (j, a[i]) to the tree, and decrement the degrees of j and a[i]. In pseudo-code:
8 for each value in do 9 for each node in do 10 if degree[] = 1 then 11 Insert edge[, ] into 12 degree[] ← degree[] - 1 13 degree[] ← degree[] - 1 14 break
At the end of this loop two nodes with degree 1 will remain (call them u, v). Lastly, add the edge (u,v) to the tree.
15 ← ← 0 16 for each node in 17 if degree[] = 1 then 18 if = 0 then 19 ← 20 else 21 ← 22 break 23 Insert edge[, ] into 24 degree[] ← degree[] - 1 25 degree[] ← degree[] - 1 26 return
Cayley's formula
The Prüfer sequence of a labeled tree on vertices is a unique sequence of length on the labels 1 to . For a given sequence of length on the labels 1 to , there is a unique labeled tree whose Prüfer sequence is .
The immediate consequence is that Prüfer sequences provide a bijection between the set of labeled trees on vertices and the set of sequences of length on the labels 1 to . The latter set has size , so the existence of this bijection proves Cayley's formula, i.e. that there are labeled trees on vertices.
Other applications Source: Cayley's formula can be strengthened to prove the following claim: The number of spanning trees in a complete graph with a degree specified for each vertex is equal to the multinomial coefficient
The proof follows by observing that in the Prüfer sequence number appears exactly times.
Cayley's formula can be generalized: a labeled tree is in fact a spanning tree of the labeled complete graph. By placing restrictions on the enumerated Prüfer sequences, similar methods can give the number of spanning trees of a complete bipartite graph. If is the complete bipartite graph with vertices 1 to in one partition and vertices to in the other partition, the number of labeled spanning trees of is , where . Generating uniformly distributed random Prüfer sequences and converting them into the corresponding trees is a straightforward method of generating uniformly distributed random labelled trees.
References
External links Prüfer code – from MathWorld