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.