The mean value of the squared path-difference distance for rooted phylogenetic trees

The path-difference metric is one of the oldest distances for the comparison of fully resolved phylogenetic trees, but its statistical properties are still quite unknown. In this paper we compute the mean value of the square of the path-difference metric between two fully resolved rooted phylogenetic trees with $n$ leaves, under the uniform distribution. This complements previous work by Steel and Penny, who computed this mean value for fully resolved unrooted phylogenetic trees.