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

The cophenetic metrics dφ,p, for p ∈ {0} ∪ [1,∞[, are a recent addition to thekit of available distances for the comparison of phylogenetic trees. Based on afifty years old idea of Sokal and Rohlf, these metrics compare phylogenetic treeson a same set of taxa by encoding them by means of their vectors of copheneticvalues of pairs of taxa and depths of single taxa, and then computing the Lnorm of the difference of the corresponding vectors. In this paper we computethe expected value of the square of dφ,2 on the space of fully resolved rootedphylogenetic trees with n leaves, under the Yule and the uniform probabilitydistributions.