An Efficient Algorithm for Generating Prüfer Codes from Labelled Trees

According to Cayley's tree formula, there are n ^n-2 labelled trees on n vertices. Prüfer gave a bijection between the set of labelled trees on n vertices and sequences of n-2 numbers, each in the range 1, 2, ..., n . Such a number sequence is called a Prüfer code. The straightforward implementation of his bijection takes O(n log n ) time. In this paper we propose an O(n) time algorithm for the same problem. Our algorithm can be easily parallelized so that a Prüfer code can be generated in O (log n ) time using O(n) processors on the EREW PRAM computational model. Received October 2, 1998, and in final form March 1, 1999.