On the number of t-ary trees with a given path length

TL;DR

This paper derives an asymptotic formula for counting t-ary trees with a fixed path length, linking combinatorics and information theory by estimating the number of possible Lempel-Ziv dictionaries for sequences.

Contribution

It provides a novel asymptotic expression for the number of t-ary trees with a given path length, connecting combinatorial enumeration with information theory applications.

Findings

Asymptotic formula for t-ary trees with path length p

Connection to the number of universal types in information theory

Estimation of Lempel-Ziv dictionaries for sequences

Abstract

We show that the number of $t$-ary trees with path length equal to $p$ is $\exp(h(t^{-1})t\log t \frac{p}{\log p}(1+o(1)))$, where $\entropy(x){=}{-}x\log x {-}(1{-}x)\log (1{-}x)$ 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 $t$.