Consider the balls $B(a,r)$ and $B(b,r)$ for the radii $r\in[0,\infty]$. Define $r_0:=|b-a|_\LL$. For $r<r_0$ the balls $B(a,r)$ and $B(b,r)$ are disjoint but for $r\geq r_0$ they are equal according to Lemma \[ball\] (cf Figure \[Berkline\]).

Hence we have to think about $\P^1 _{Berk}$ as real lines attached to the elements of $\LL\cup\{\infty\}$, identified to the balls of radius $0$ or infinity, parametrized by the radius of the balls and with intersecting points at each values of $|a-b|_\LL$ for some $a$ and $b$ in $\LL$ (cf Figure \[P1brek\]).

Proposition \[prethm\] gives a good idea of what is the all space. For the purpose of proving that trees of spheres are included in the Berkovich space, it will be sufficient to consider the elements mentioned in this statement. We will give in Section \[Chap7\] a complete version of this theorem.The points that corresponds to balls of radius 0 are identified to elements of $\P^1\LL$ and are called type I. The non branching points are said to be type II and the branching ones are type III. The points that we omitting are some ends of the Berkovich tree. These points are called the type IV points. We will totally abuse and design 