Prüfer sequence

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 n vertices has length n − 2, and can be generated by a simple iterative algorithm. Prüfer sequences were first used by Heinz Prüfer to prove Cayley's formula in 1918.[1]

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 T with vertices {1, 2, ..., n}. At step i, remove the leaf with the smallest label and set the ith 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 n − 2.

Example

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 {a[1], a[2], ..., 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(a)
 1 n ← length[a]
 2 T ← a graph with n + 2 isolated nodes, numbered 1 to n + 2
 3 degree ← an array of integers
 4 for each node i in T do
 5     degree[i] ← 1
 6 for each value i in a do
 7     degree[i] ← degree[i] + 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 i in a do
 9     for each node j in T do
10         if degree[j] = 1 then
11             Insert edge[i, j] into T
12             degree[i] ← degree[i] - 1
13             degree[j] ← degree[j] - 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.[2]

15 u ← v ← 0
16 for each node i in T
17     if degree[i] = 1 then
18         if u = 0 then
19              u ← i
20         else
21              v ← i
22              break
23 Insert edge[u, v] into T
24 degree[u] ← degree[u] - 1
25 degree[v] ← degree[v] - 1
26 return T

Cayley's formula

The Prüfer sequence of a labeled tree on n vertices is a unique sequence of length n − 2 on the labels 1 to n. For a given sequence S of length n–2 on the labels 1 to n, there is a unique labeled tree whose Prüfer sequence is S.

The immediate consequence is that Prüfer sequences provide a bijection between the set of labeled trees on n vertices and the set of sequences of length n − 2 on the labels 1 to n. The latter set has size nⁿ⁻², so the existence of this bijection proves Cayley's formula, i.e. that there are nⁿ⁻² labeled trees on n vertices.

Other applications[3]

Cayley's formula can be strengthened to prove the following claim:

The number of spanning trees in a complete graph

K_{n}

with a degree

d_{i}

specified for each vertex

i

is equal to the multinomial coefficient

{\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)!}}.

The proof follows by observing that in the Prüfer sequence number

i

appears exactly

(d_{i}-1)

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 G is the complete bipartite graph with vertices 1 to n₁ in one partition and vertices n₁ + 1 to n in the other partition, the number of labeled spanning trees of G is $n_{1}^{n_{2}-1}n_{2}^{n_{1}-1}$ , where n₂ = n − n₁.
Generating uniformly distributed random Prüfer sequences and converting them into the corresponding trees is a straightforward method of generating uniformly distributed random labelled trees.

gollark: I didn't check.

gollark: I used to use that, but accidentally erased my knowledge of category theory.

gollark: Oh, yes.

gollark: Just got to execute infinitely many operations in finite time.

gollark: It's really convenient, I must say.

References

Prüfer, H. (1918). "Neuer Beweis eines Satzes über Permutationen". Arch. Math. Phys. 27: 742–744.
Jens Gottlieb; Bryant A. Julstrom; Günther R. Raidl; Franz Rothlauf. (2001). "Prüfer numbers: A poor representation of spanning trees for evolutionary search" (PDF). Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2001): 343–350. Archived from the original (PDF) on 2006-09-26.
Kajimoto, H. (2003). "An Extension of the Prüfer Code and Assembly of Connected Graphs from Their Blocks". Graphs and Combinatorics. 19: 231–239. doi:10.1007/s00373-002-0499-3.

External links

Prüfer code – from MathWorld

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[1] Prüfer, H. (1918). "Neuer Beweis eines Satzes über Permutationen". Arch. Math. Phys. 27: 742–744.

[2] Jens Gottlieb; Bryant A. Julstrom; Günther R. Raidl; Franz Rothlauf. (2001). "Prüfer numbers: A poor representation of spanning trees for evolutionary search" (PDF). Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2001): 343–350. Archived from the original (PDF) on 2006-09-26.

[3] Kajimoto, H. (2003). "An Extension of the Prüfer Code and Assembly of Connected Graphs from Their Blocks". Graphs and Combinatorics. 19: 231–239. doi:10.1007/s00373-002-0499-3.