\zeta$ is a branching point separating $a\approx B(a,0)$ and $b\approx B(b,0)$. Thus $a$ and $b$ are in the same branch if and only if $|b-a|_\LL<1$. The result follows by remarking that $|b-a|_\LL<1$ if and only if the corresponding series have the same constant term.

We denote the set of branches at $\zeta$ by $T_{\zeta} \P^1 _{Berk}$. It is usually called the tangent space at $\zeta$. More generally, for any branching point $a\in\P^1_{Berk}$, we denote $T_{\zeta} \P^1 _{Berk}$ its tangent space, ie the set of branches at $a$.

\[remopenball\] Note that the intersection every branch at a branching point $B(a,r)$ (for $a\in\P^1_\LL$ and $r>0$) with $\P^1\LL$ is empty or an open ball. Indeed, after a change of coordinate we can suppose that this branch $B$ does not contains $\infty\in\P^1\LL$. Take $a'\in B\cap\LL$. Then $B(a,r)\cap\LL$ is the increasing union of the closed balls $B(a',r')$ for $0\leq r'<r$.

Comparison {#compspace}
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In order to be able to compare these two worlds, we have to first consider families of marked spheres instead of sequences. We suppose that for all $t\in D(0,1)\setminus \{0\}$, the maps $i_t:X\to\S_t$ are injections from the finite set $X$ to some spheres that we identify to the Riemann sphere $\C\cup\{\infty\}$. For 