  The reconstruction of a central tendency `species tree' from a large number
of conflicting gene trees is a central problem in systematic biology. Moreover,
it becomes particularly problematic when taxon coverage is patchy, so that not
all taxa are present in every gene tree. Here, we list four apparently
desirable properties that a method for estimating a species tree from gene
trees could have (the strongest property states that building a species tree
from input gene trees and then pruning leaves gives a tree that is the same as,
or more resolved than, the tree obtained by first removing the taxa from the
input trees and then building the species tree). We show that while it is
technically possible to simultaneously satisfy these properties when taxon
coverage is complete, they cannot all be satisfied in the more general
supertree setting. In part two, we discuss a concordance-based consensus method
based on Baum's `plurality clusters', and an extension to concordance
supertrees.
