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| >[(n 􀀀 3)/(r - 1)], 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 [(n - 3)/3] that defines T , but there is a phylogenetic X-tree with n = 12 which is not de ned by any set of 3 characters.