The Number of Huffman Codes, Compact Trees, and Sums of Unit Fractions

The number of “nonequivalent” compact 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.