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

[[File:Tree graph.svg|right|frame|A labeled tree with Prüfer sequence [4,4,4,5].]]

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 <code>[a[1], a[2], ..., a[n]]</code> be a Prüfer sequence:

The tree will have <code>n+2</code> nodes, numbered from <code>1</code> to <code>n+2</code>.

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 <code>a[i]</code>, find the first (lowest-numbered) node, <code>j</code>, with degree equal to 1, add the edge <code>(j, a[i])</code> to the tree, and decrement the degrees of <code>j</code> and <code>a[i]</code>. 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 <code>u</code>, <code>v</code>). Lastly, add the edge <code>(u,v)</code> 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&nbsp;. For a given sequence of length on the labels 1 to&nbsp;, there is a unique labeled tree whose Prüfer sequence is&nbsp;.

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 <math>K_n</math> with a degree <math>d_i</math> specified for each vertex <math>i</math> is equal to the multinomial coefficient

::<math>\binom{n-2}{d_1-1,\,d_2-1,\,\dots,\,d_n-1}=\frac{(n-2)!}{(d_1-1)!(d_2-1)!\cdots(d_{n}-1)!}.</math>

:The proof follows by observing that in the Prüfer sequence number <math>i</math> appears exactly <math display=inline>d_i-1</math> 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 <math>n_1^{n_2-1} n_2^{n_1-1}</math>, 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

  • Prüfer code – from MathWorld