In a paper with Billera and Holmes published in 2001 we studied the space of finite metric trees with a fixed number of leaves, which can be interpreted as phylogenetic trees with positive branch lengths. That paper included some information about how to find the geodesic path between two trees and the corresponding distance, but did not include a specific algorithm for doing so.
The current notes address this task. They grew out of conversations with John Smillie, and were written shortly after the Billera-Holmes-Vogtmann paper appeared. They were never published, partly because I believe that a little more work (which I haven't done) could produce a substantially better algorithm. Meanwhile several people have requested these notes, so I decided to make them generally available.