Capturing a phylogenetic tree when the number of character states varies with the number of leaves

TL;DR

This paper proves that for large enough phylogenetic trees, it is possible to select a small set of characters with limited states that uniquely determine the tree, using probabilistic methods.

Contribution

It establishes a new combinatorial result showing the existence of small character sets with bounded states that capture any large binary phylogenetic tree.

Findings

Existence of character sets with size proportional to n^α

Characters can have at most n^β states

Unique tree capture guaranteed for large n

Abstract

We show that for any two values $\alpha, \beta >0 $ for which $\alpha+\beta>1$ then there is a value $N$ so that for all $n \geq N$ the following holds. For any binary phylogenetic tree $T$ on $n$ leaves there is a set of $\lfloor n^\alpha \rfloor$ characters that capture $T$, and for which each character takes at most $\lfloor n^\beta \rfloor$ distinct states. Here `capture' means that $T$ is the unique perfect phylogeny for these characters. Our short proof of this combinatorial result is based on the probabilistic method.