# 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 *i*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 *n* − 2.

### 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[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) 1n←length[a] 2T← a graph withn+ 2 isolated nodes, numbered 1ton+ 2 3degree← an array of integers 4foreach nodeiinT5dodegree[i] ← 1 6foreach valueiina7dodegree[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:

8foreach valueiina9foreach nodejinT10ifdegree[j] = 1 11thenInsertedge[i,j] intoT12degree[i] ←degree[i] - 1 13degree[j] ←degree[j] - 1 14break

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]}

14u←v← 0 15foreach nodeiinT16ifdegree[i] = 1 17thenifu= 0 18thenu←i19elsev←i20break21 Insertedge[u,v] intoT22degree[u] ←degree[u] - 1 23degree[v] ←degree[v] - 1 24returnT

## 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 *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 *n*^{n−2}, so the existence of this bijection proves Cayley's formula, i.e. that there are
*n*^{n−2} labeled trees on *n* vertices.

## Other applications

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

- The number of spanning trees in a complete graph with degrees 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
*G*is the complete bipartite graph with vertices 1 to*n*_{1}in one partition and vertices*n*_{1}+ 1 to*n*in the other partition, the number of labeled spanning trees of*G*is , where*n*_{2}=*n*−*n*_{1}.

- 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