A central problem that arises in evolutionary biology is that of displaying partitions of subsets of a finite set X on a tree whose vertices are partially labelled with the elements of X. Such a tree is called an X-tree and, for a collection C of partitions of subsets of X, characterisations for the existence and uniqueness of an X-tree that displays C have been previously given in terms of cho...

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 ther...

A phylogenetic tree on a label set L is a tree which exactly ILl leaves (vert.ices of degree one), no vertices of degree two, and each leaf labeled with a distinct element from L. A binary phylogenetic tree is a phylogenetic tree in which every non-leaf has degree three. Phylogenetic trees get their name because they are often appropriate models for evolutionary history. However, for simplicity...

A fundamental problem in the study of phylogenetic networks is to determine whether or not a given phylogenetic network contains a given phylogenetic tree. We develop a quadratic-time algorithm for this problem for binary nearly-stable phylogenetic networks. We also show that the number of reticulations in a reticulation visible or nearly stable phylogenetic network is bounded from above by a f...