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

The cophenetic metrics d_φ,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 L^p norm of the difference of the corresponding vectors. In this paper we compute the expected value of the square of d_φ,2 on the space of fully resolved rooted phylogenetic trees with n leaves, under the Yule and the uniform probability distributions.