internal vertex $v$ of $T^X$,

    -   a Riemann sphere $\S_v$ and

    -   a one-to-one map $i_v:E_v\to \S_v$.

We denote by $\overline{\rm Mod}_X$ the space of trees of spheres marked by $X$. For $e\in E_v$, we say that $i_v(e)$ is the attaching point of $e$ on $v$. We extend $i_v$ to $T\setminus \{v\}$ by setting $i_v(v'):=i_v(e)$ if $v'\in B_v(e)$. We denote by $a_v$ the application $i_v|_X$. When the tree has a unique internal vertex $v$, we identify it to the marked sphere $a_v$.

A sequence of marked spheres $a_n:X\to{\mathbb S}_n$ converges to a tree of sphere ${\cal T}^X$ if for all internal vertex $v$ of $\T^X$ , there exists an isomorphism $M_{n,v}:{\mathbb S}_n\to{\S}_v$ such that $M_{n,v} \circ a_n$ converges to $a_v$.

(We prefer to use the notation $\S_n$ instead of $\S$ or $\P^1\C$ because the $\S_n$ should be thought distinct.)

We keep in this paper this notion of convergence witch is not Hausdorff. In fact this notion is Hausdorff after passing to a quotient by the appropriated notion of isomorphism of trees of spheres. We will not develop this notion of isomorphism in this paper but the interested reader can take a look to [@A3].

Berkovich space
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Recall that $\LL$ 