{\mathbf}R$ (e.g. the value group of $K$), we say that $\Gamma$ is a *$\Lambda$-metric graph* if it admits a graph model $G = (V, E)$ where the weight of every finite edge of $G$ belongs to $\Lambda$.

Given a graph model $G = (V, E)$ for $\Gamma$, the *$\Lambda$-rational points* of $\Gamma$ are the points whose distance to some (and hence every) vertex is an element of $\Lambda$— we call this set $\Gamma(\Lambda)$.

See Section 2.1 of [@ABBR] for another description of a metric graph.

We recall that a *spanning tree* of a (connected) graph is a maximal, acyclic collection of edges such that every vertex of the graph is an endpoint of one of these edges. If $e_1,\dots,e_g$ form the complement of such a spanning tree, then $g$—which is well defined—is called the *genus* of $G$. One can check that if $G$ is a graph model of $\Gamma$ then $g = \dim_{{\mathbf}Q} {\rm H}_1(\Gamma;{\mathbf}Q)$.

Berkovich analytic spaces
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For every variety $X$ over $K$, there is a topological space, ${X^{\rm an}}$, introduced by Berkovich [@Berk] called the *Berkovich analytification*. The points of ${X^{\rm an}}$ are pairs $(p_x, |\cdot|_x)$ where $p_x \in X$ and $|\cdot|_x$ is an absolute value on 