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

Phylogenetic trees are trees whose leaves are bijectively labelled by a set of species, or more generally, taxa. A character is an assignment of states to the taxa, where each state represents an evolutionary trait, and a character is said to be convex on a tree if the subtrees spanned by the states are disjoint. In this article we prove that the maximum parsimony distance d<inf>MP</inf>(T<inf>1</inf>,T<inf>2</inf>)=k between two unrooted binary phylogenetic trees T<inf>1</inf>, T<inf>2</inf>, which informally seeks a character that can be explained by one tree much better than the other, can be defined by a character that is convex on one of T<inf>1</inf>, T<inf>2</inf> 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 = 1 there exist two trees T<inf>1</inf>, T<inf>2</inf> with d<inf>MP</inf>(T<inf>1</inf>,T<inf>2</inf>)=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<inf>1</inf>, T<inf>2</inf>. 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.