  We study the behavior of phylogenetic tree shapes in the tropical geometric
interpretation of tree space. Tree shapes are formally referred to as tree
topologies; a tree topology can also be thought of as a tree combinatorial
type, which is given by the tree's branching configuration and leaf labeling.
We use the tropical line segment as a framework to define notions of variance
as well as invariance of tree topologies: we provide a combinatorial search
theorem that describes all tree topologies occurring along a tropical line
segment, as well as a setting under which tree topologies do not change along a
tropical line segment. Our study is motivated by comparison to the moduli space
endowed with a geodesic metric proposed by Billera, Holmes, and Vogtmann
(referred to as BHV space); we consider the tropical geometric setting as an
alternative framework to BHV space for sets of phylogenetic trees. We give an
algorithm to compute tropical line segments which is lower in computational
complexity than the fastest method currently available for BHV geodesics and
show that its trajectory behaves more subtly: while the BHV geodesic traverses
the origin for vastly different tree topologies, the tropical line segment
bypasses it.
