Efficient Prüfer-Like Coding and Counting Labelled Hypertrees

Abstract

We consider labelled r-uniform hypertrees, 2≤r≤n, where n is the number of vertices in the hypertree. Any two hyperedges in a hypertree share at most one vertex and each hyperedge in an r-uniform hypertree contains exactly r vertices. We show that r-uniform hypertrees can be encoded in linear time using as little as n−2 integers in the range [1,n]. The decoding algorithm also runs in linear time. For general hypertrees, we require codes of length n+p−2 where p is the number of vertices belonging to more than one hyperedge in the given hypertree. Based on our coding technique, we show that there are at most $\frac {n^{(n-2)}-f(n,r)}{(r-1)^{(r-2)*\frac{n-1}{r-1}}}$ distinct labelled r-uniform hypertrees, where f(n,r) is a lower bound on the number of labelled trees with maximum (vertex) degree exceeding $(r-1)+\frac {n-1}{r-1}-2$ . We suggest a counting scheme for determining such a lower bound f(n,r).