Preserving consensus hierarchies

In numerical taxonomy we often have the task of finding a consensus hierarchy for a given set of hierarchies. This consensus hierarchy should reflect the substructures which are common to all hierarchies of the set. Because there are several kinds of substructures in a hierarchy, the general axiom to preserve common substructures leads to different axioms for each kind of substructure. In this paper we consider the three substructurescluster, separation, andnesting, and we give several characterizations of hierarchies preserving these substructures. These characterizations facilitate interpretation of axioms for preserving substructures and the examination of properties of consensus methods. Finally some extensions concerning the preserving of qualified substructures are discussed.The author is grateful to the editor and the referees for their helpful suggestions and to H. J. Bandelt for his comments on an earlier version of this paper.