Some axiomatic limitations for consensus and supertree functions on hierarchies

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.