  Tree comparison metrics have proven to be an invaluable aide in the
reconstruction and analysis of phylogenetic (evolutionary) trees. The
path-length distance between trees is a particularly attractive measure as it
reflects differences in tree shape as well as differences between branch
lengths. The distance equals the sum, over all pairs of taxa, of the squared
differences between the lengths of the unique path connecting them in each
tree. We describe an $O(n \log n)$ time for computing this distance, making
extensive use of tree decomposition techniques introduced by Brodal et al.
(2004).
