Note on expected internode distances for gene trees in species trees

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.