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

The number of “nonequivalent” compact Huffman codes of length <i>r</i> over an alphabet of size <i>t</i> has been studied frequently. Equivalently, the number of “nonequivalent” complete <i>t</i>-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.

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