The number of Huffman codes, compact trees, and sums of unit fractions

TL;DR

This paper surveys existing research on counting nonequivalent Huffman codes and complete t-ary trees, and provides a precise asymptotic formula for their enumeration, improving previous results.

Contribution

It unifies various approaches to counting Huffman codes and trees, and derives a more accurate asymptotic expression for their enumeration.

Findings

Derived a precise asymptotic formula with main terms and error bounds

Unified multiple independent approaches in the literature

Enhanced understanding of the enumeration of Huffman codes and t-ary trees

Abstract

The number of "nonequivalent" Huffman codes of length r over an alphabet of size t has been studied frequently. Equivalently, the number of "nonequivalent" complete t-ary trees has been examined. We first survey the literature, unifying several independent approaches to the problem. Then, improving on earlier work we prove a very precise asymptotic result on the counting function, consisting of two main terms and an error term.