Defining a Phylogenetic Tree with the Minimum Number of r-State Characters

Semple and Steel (2002) showed that if T is a phylogenetic X-tree and C is a collection of r-state characters that defines T , then |C| ≥ d(n− 3)/(r − 1)e, where n = |X|. In this paper, we show that, provided n is sufficiently large, this lower bound is sharp. Furthermore, we show that, for all n ≥ 13, there exists a collection of 4-state characters of size d(n − 3)/3e that defines T , but there is a phylogenetic X-tree with n = 12 which is not defined by any set of 3 characters.