Trees, Taxonomy, and Strongly Compatible Multi-state Characters

Given a family ofbinarycharacters defined on a setX, a problem arising in biological and linguistic classification is to decide whether there is a tree structure onXwhich is “compatible” with this family. A fundamental result from hierarchical clustering theory states that there exists a tree structure onXfor such a family if and only if any two of the characters arecompatible. In this paper, we prove a generalization of this result. Namely, we show that given a family ofmulti-statecharacters onXwhich we denote by ?, there exists a tree structure onX, called an (X,?)-tree, which is “compatible” with ? if and only if any two of the characters arestrongly compatible. To prove this result, we introduce the concept ofblock systems, set theoretical structures which arise naturally from, amongst other things,block graphs, and the related concepts ofblock interval systemsand ?-systems.