Improved Encoding and Counting of Uniform Hypertrees

TL;DR

This paper derives an exact count of labeled r-uniform hypertrees with n vertices and introduces an efficient encoding scheme that approaches the theoretical minimum number of bits needed for such structures.

Contribution

It provides a closed-form formula for counting labeled r-uniform hypertrees and proposes an encoding method close to optimal in information-theoretic terms.

Findings

Exact enumeration formula for labeled r-uniform hypertrees.

Encoding scheme with minimal overhead over entropy bound.

Efficient representation of hypertrees in terms of bits.

Abstract

We consider labeled $r$-uniform hypertrees having $n \ge r \ge 2$ vertices. The number of hyperedges in such a hypertree is $m = (n - 1)/(r - 1)$. We show that there are exactly $f(n, r) = \frac{(n-1)! n^{m-1}}{(r-1)!^m m!}$ $r$-uniform hypertrees with $n$ vertices labeled with distinct integers. We also give an encoding scheme that encodes such hypertrees using, on an average, at most $1 + \log_2 e$ bits more than $\log_2(f(n, r))$.