A Linear Bound on the Number of States in Optimal Convex Characters for Maximum Parsimony Distance

Given two phylogenetic trees on the same set of taxa X, the maximum parsimony distance d(MP) is defined as the maximum, ranging over all characters chi on X, of the absolute difference in parsimony score induced by chi on the two trees. In this note, we prove that for binary trees there exists a character achieving this maximum that is convex on one of the trees (i.e., the parsimony score induced on that tree is equal to the number of states in the character minus 1) and such that the number of states in the character is at most 7d(MP) - 5. This is the first non-trivial bound on the number of states required by optimal characters, convex or otherwise. The result potentially has algorithmic significance because, unlike general characters, convex characters with a bounded number of states can be enumerated in polynomial time.