The expected value of the squared cophenetic metric under the Yule and the uniform models

The cophenetic metrics dφ,p, for p ∈ {0} ∪ [1, ∞), are a recent addition to the kit of available distances for the comparison of phylogenetic trees. Based on a fifty years old idea of Sokal and Rohlf, these metrics compare phylogenetic trees on a same set of taxa by encoding them by means of their vectors of cophenetic values of pairs of taxa and depths of single taxa, and then computing the Lp norm of the difference of the corresponding vectors. In this paper we compute the expected value of the square of dφ,2 on the space of fully resolved rooted phylogenetic trees with n leaves, under the Yule and the uniform probability distributions.