  In a recent paper on 'Estimating Species Trees from Unrooted Gene Trees' Liu
and Yu observe that the distance matrix on the underlying taxon set, which is
built up from expected internode distances on gene trees under the multispecies
coalescent, is tree-like, and that the underlying additive tree has the same
topology as the true species tree. Hence they suggest to use (observed) average
internode distances on gene trees as an input for the neighbor joining
algorithm to estimate the underlying species tree in a statistically consistent
way. In this note we give a rigorous proof of their above mentioned
observation.
