Classifying Tree Topologies along Tropical Line Segments

The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or "four clade rearrangement" moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as $n^2$, but the average number of moves when the two endpoint trees are chosen at random is $O(n (\log n)^4)$. This is in contrast with $O(n \log n)$, the average number of NNI moves needed to transform one tree into another.