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

The cophenetic metrics $d_{\varphi,p}$, for $p\in {0}\cup[1,\infty[$, 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 $d_{\varphi,2}$ on the space of fully resolved rooted phylogenetic trees with $n$ leaves, under the Yule and the uniform probability distributions.