  The number of the non-shared edges of two phylogenies is a basic measure of
the dissimilarity between the phylogenies. The non-shared edges are also the
building block for approximating a more sophisticated metric called the nearest
neighbor interchange (NNI) distance. In this paper, we give the first
subquadratic-time algorithm for finding the non-shared edges, which are then
used to speed up the existing approximating algorithm for the NNI distance from
$O(n^2)$ time to $O(n \log n)$ time. Another popular distance metric for
phylogenies is the subtree transfer (STT) distance. Previous work on computing
the STT distance considered degree-3 trees only. We give an approximation
algorithm for the STT distance for degree-$d$ trees with arbitrary $d$ and with
generalized STT operations.
