The expected value under the Yule model of the squared path-difference distance

The path-difference metric is one of the oldest and most popular distances for the comparison of phylogenetic trees, but its statistical properties are still quite unknown. In this work we compute the expected value under the Yule model of evolution of its square on the space of fully resolved rooted phylogenetic trees with n leaves. This complements previous work by Steel and Penny and by Mir and Rossellَ, who computed this mean value for fully resolved unrooted and rooted phylogenetic trees, respectively, under the uniform distribution.