that these infinite edges be leaf edges.

Explicitly, there exists a set of vertices $V$ and set of edges $E$. Every edge $e$ has a distance function $\operatorname{d}_e$, such that $e$ is isometric to a closed interval. Finally, there are maps $\partial e \to V$ that tell us how to glue the edges to the vertices. Every edge of $\Gamma$ has the usual distance function which we extend to $\Gamma$ by setting $\operatorname{d}(x,y) =$ the length of the shortest path from $x$ to $y$.

A choice of $G = (V, E)$ is called a *graph model* of $\Gamma$. We forget about all the distance functions and topologies on $E$ and just remember the lengths. In this way, $G$ is a graph where each edge $e \cong [0,l]$ has an associated length $l$. If $G$ is a graph model of $\Gamma$, then so is any length-respecting subdivision of $G$. When the graph model is fixed, we may refer to edges and vertices of $G$ as edges and vertices of $\Gamma$.

Usually one would call $G$ a “weighted graph” but since the term “weight” is used in relation to the tropical balancing condition, we avoid this here.

Given a subgroup $\Lambda \subseteq 