We know that the inverse image of any vertice $v$ is the closed unit disc punctured by finitely many maximal open rational discs and the number of these maximal open discs corresponds to the index of $v$ and it is equal to $g+1$ since $T(\Gamma)$ is the universel covering of $T(\Gamma)/\Gamma$ and $T(\Gamma)/\Gamma$ has $g+1$ edges.

$T(\Gamma)$ is constructed by gluing semi-stable skeletons and hence the inverse image of any vertice $v$ is the closed unit disc punctured by finitely many maximal open rational discs and the number of these maximal open discs corresponds to the index of $v$ and it is equal to $g+1$ since $T(\Gamma)$ is the universel covering of $T(\Gamma)/\Gamma$ and $T(\Gamma)/\Gamma$ has $g+1$ edges.