A strengthened bound on the number of states required to characterize maximum parsimony distance

TL;DR

This paper proves a tighter bound on the number of states needed to characterize maximum parsimony distance between two phylogenetic trees, improving previous bounds and providing empirical evidence of fewer states being needed in practice.

Contribution

It establishes a new upper bound of 2k states for characters convex on one tree, improving the previous bound of 7k-5, and demonstrates that fewer states are often sufficient empirically.

Findings

New bound of 2k states for maximum parsimony distance

Existence of trees requiring at least k+1 states

Empirical analysis shows fewer than k+1 states are usually needed

Abstract

In this article we prove that the distance $d_{\mathrm{MP}}(T_1,T_2) = k$ between two unrooted binary phylogenetic trees $T_1, T_2$ on the same set of taxa can be defined by a character that is convex on one of $T_1, T_2$ and which has at most $2k$ states. This significantly improves upon the previous bound of $7k-5$ states. We also show that for every $k \geq 1$ there exist two trees $T_1, T_2$ with $d_{\mathrm{MP}}(T_1,T_2) = k$ such that at least $k+1$ states are necessary in any character that achieves this distance and which is convex on one of $T_1, T_2$. We augment these lower and upper bounds with an empirical analysis which shows that in practice significantly fewer than $k+1$ states are usually required.