every $a\in X$ we denote by $a_t$ the element $i_t(a)$. We suppose that the $a_t$ depend holomorphicaly on $t$.

Then naturally, every $a_t$ for $a\in X$ is identified to an element of $\LL$. Now take two other elements $b$ and $c$ in $X$ and identify them as for $a$ to elements $b_t$ and $c_t$ in $\LL$. Note that $a_t, b_t$ and $c_t$ are also identified to ends of the tree $\P^1_{Berk}$, thus there is a unique branching point $v\in \P^1_{Berk}$ separating them. Denote by $\S_v$ the tangent space $T_v\P^1_{Berk}$. Applying a Moebius transformation $M_{v,t}$ with coefficients in $\LL$ we can suppose that $a_t=1$, $b_t=1$ and $c_t=\infty$. Then $v=\zeta$ so we can deduce that $\rho(a_t)=\lim_{t\to0}a_t$. It follows that two elements of $M_{v,t}\circ i_t(X)$ have the same limit on $\S_v$ if and only if they are in the same branch of $\S_v$.

Every sphere $\S_v$ in $\T^X$ separates at least three elements of $X$, so for each sphere we can associate a vertex $v$ in $\P^1_{Berk}$ and its tangent space to $\S_v$ and the notions of convergence corresponds: for every vertex in $\P^1_{Berk}$ corresponding to a vertex in $\T^X$, there is a Moebius transformation $M_t$ such that $a$ and $b\in X$ are 