  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.
