On the Number of t-Ary Trees with a Given Path Length

We show that the number of t-ary trees with path length equal to p is t h(t-1) tp log2 p(1+o(1)) , where h(x)=-x log2 x-(1-x) log2(1-x) is the binary entropy function. Besides its intrinsic combinatorial interest, the question recently arose in the context of information theory, where the number of t-ary trees with path length p estimates the number of universal types, or, equivalently, the number of different possible Lempel-Ziv'78 dictionaries for sequences of length p over an alphabet of size t.

• Type: Journal
• Year: 2006
• Where: ALGORITHMICA
• Authors: Gadiel Seroussi
• Keywords: ALGORITHMICA 2006, Binary Entropy Function, Path Length, T-ary Trees