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.


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:

 1 nlength[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
 5     do degree[i] ← 1
 6 for each value i in a
 7     do 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
 9     for each node j in T
10          if degree[j] = 1
11             then 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 uv ← 0
16 for each node i in T
17     if degree[i] = 1
18         then if u = 0
19             then ui
20             else vi
21                  break
22 Insert edge[u, v] into T
23 degree[u] ← degree[u] - 1
24 degree[v] ← degree[v] - 1
25 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 this much is clear. Somewhat less obvious is the fact that for a given sequence S of length n2 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 n2 on the labels 1 to n. The latter set has size nn2, so the existence of this bijection proves Cayley's formula, i.e. that there are nn2 labeled trees on n vertices.

Other applications[3]

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.


  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, and 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.
  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.

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