  Tropical geometry with the max-plus algebra has been applied to statistical
learning models over tree spaces because geometry with the tropical metric over
tree spaces have some nice properties. One of the challenges in applications of
tropical geometry to tree spaces is the difficulty to interpret outcomes of
statistical models with the tropical metric. This paper focuses on
combinatorics of tree topologies along a tropical line segment, an intrinsic
geodesic with the tropical metric, between two phylogenetic trees over the tree
space and we show some properties of a tropical line segment between two trees.
Especially we show that a probability of a tropical line segment of randomly
chosen two trees going thorough the origin (the star tree) is zero and we also
show that if two given trees differ only one nearest neighbor interchange (NNI)
move, then the tree topology of a tree in the tropical line segment between
them is the same tree topology of one of these given two trees with possible
zero branch lengths.
