  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.
