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Consensus trees and supertrees are regularly used in systematic biology in order to obtain a summary for the common agreement of the evolutionary relationships among a collection of phylogenetic trees (hierarchies). When every tree is defined on the same set of taxa then consensus functions are used, while if the trees are defined on different sets then supertree functions are used. For both of these situations we will consider some of the limitations that might arise from the placing of singularly reasonable and apparently innocuous conditions on the functions. Previous work is reviewed together with new material. In particular, we consider the impact of axioms requiring that the removal or addition of a tree that contains no, or no new, branching information should not affect the outcome. [ABSTRACT FROM AUTHOR]

Abstract: A median of a sequence of elements of a finite metric space is an element x for which is minimum. The function M with domain the set of all finite sequences on X and defined by is a median of is called the median function on X, and is one of the most studied consensus functions. Based on previous characterizations of median sets , a generalization of the median function is introduced and studied on various graphs and ordered sets. In addition, new results are presented for median graphs. [Copyright &y& Elsevier]

Abstract: Consensus trees have been used routinely in systematic biology for over 30 years with the majority-rule consensus tree among the most popular. In this paper we give axioms that characterize a new consensus function, called majority-rule (+), that is based on a generalization of majority-rule to the supertree case. In addition, two other related consensus functions are characterized. [Copyright &y& Elsevier]