A scale-free method for testing the proportionality of branch lengths between two phylogenetic trees

We introduce a scale-free method for testing the proportionality of branch lengths between two phylogenetic trees that have the same topology and contain the same set of taxa. This method scales both trees to a total length of 1 and sums up the differences for each branch. Compared to previous methods, ours yields a fully symmetrical score that measures proportionality without being affected by scale. We call this score the normalized tree distance (NTD). Based on real data, we demonstrate that NTD scores are distributed unimodally, in a manner similar to a lognormal distribution. The NTD score can be used to, for example, detect co-evolutionary processes and measure the accuracy of branch length estimates.