isomorphism.

Limits and trees of spheres
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As we saw above, the structure of the trees we are going to define is essentially combinatorial. It is important as it encodes a rigidity that is not obvious to describe with a simple vocabulary. That’s why we define everything in order to make easy to talk about it.

A graph is the disjoint union of a finite set $V$ called set of vertices and an other finite set $E$ consisting of elements of the form $\{v,v'\}$ with different $v,v'\in V$. We say that $\{ v,v'\}$ is an edge between $v$ and $v'$. For all $v\in V$ we define $E_v$ the set of edges containing $v$. We call valence of $v$ the cardinal of $E_v$.

A (stable) tree is a connected graph without cycle and whose internal vertices have at least valence $3$.

For every vertex $v$ and every edge $e$ adjacent to $v$, we denote by $B_v(e)$ the branch on $v$ that contains $e$, ie the connected component of $T\setminus\{v\}$ that contains $e$.

A tree of sphere ${\cal T^X}$ (marked by $X$) is the data of:

-   a combinatorial tree $T^X$ whose leaves are the elements of $X$ and

-   for every 