  We answer two questions raised by Bryant, Francis and Steel in their work on
consensus methods in phylogenetics. Consensus methods apply to every practical
instance where it is desired to aggregate a set of given phylogenetic trees
(say, gene evolution trees) into a resulting, "consensus" tree (say, a species
tree). Various stability criteria have been explored in this context, seeking
to model desirable consistency properties of consensus methods as the
experimental data are updated (e.g., more taxa, or more trees, are mapped).
However, such stability conditions can be incompatible with some basic
regularity properties that are widely accepted to be essential in any
meaningful consensus method. Here, we prove that such an incompatibility does
arise in the case of extension stability on binary trees and in the case of
associative stability. Our methods combine general theoretical considerations
with the use of computer programs tailored to the given stability requirements.
