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-HolmesVogtmann 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.