  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 has some nice properties such as convexity in terms of the tropical
metric. One of the challenges in applications of tropical geometry to tree
spaces is the difficulty interpreting 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. Specifically we show that a
probability of a tropical line segment of two randomly chosen trees going
through the origin (the star tree) is zero if the number of leave is greater
than four, 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.
