Prüfer sequence

The Prüfer sequence of a labeled tree is unique and has length n 2.

The proof follows by observing that in the Prüfer sequence number appears exactly times.

Prüfer sequence, giving a proof of Cayley's formula for the number of labeled trees.

Prüfer sequences yield a bijective proof of Cayley's formula.

Let be a Prüfer sequence:

Prüfer sequence (also known as Prüfer code).

Initially, vertex 1 is the leaf with the smallest label, so it is removed first and 4 is put in the Prüfer sequence.

One can generate a labeled tree's Prüfer sequence by iteratively removing vertices from the tree until only two vertices remain.

By placing restrictions on the enumerated Prüfer sequences, similar methods can give the number of spanning trees of a complete bipartite graph.

Prüfer sequences were first used by Heinz Prüfer to prove Cayley's formula in 1918.

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.

In combinatorial mathematics, the Prüfer sequence (also Prüfer code or Prüfer numbers) of a labeled tree is a unique sequence associated with the tree.

Generating uniformly distributed random Prüfer sequences and converting them into the corresponding trees is a straightforward method of generating uniformly distributed random labelled trees.

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.

As with any tree, stars may be encoded by a Prüfer sequence; the Prüfer sequence for a star K consists of k 1 copies of the center vertex.

Any tree can be uniquely encoded into a Prüfer sequence, and any Prüfer sequence can be uniquely decoded into a tree; these two results together provide a bijective proof of Cayley's formula.