mass can be lost within a sudden break: this happens as soon as $\nu (\sum_{i}s_{i}<1)\neq 0$, but we will not be interested in this phenomenon here either. The loss of mass phenomenon stated above is completely different from erosion or sudden loss of mass: it is due to the fact that small fragments tend to decay faster when $\alpha <0$.

On the other hand, let us define the notion of CRT. An $\mathbb{R}$-tree (with the terminology of Dress and Terhalle [@drter]; it is called *continuum tree set* in Aldous [@aldouscrt93]) is a complete metric space $(T,d)$, whose elements are called *vertices*, which satisfies the following two properties:

-   For $v,w\in T$, there exists a unique geodesic $[[v,w]]$ going from $v $ to $w$, i.e. there exists a unique isomorphism $\varphi
    _{v,w}:[0,d(v,w)]\rightarrow T$ with $\varphi _{v,w}(0)=v$ and $\varphi
    _{v,w}(d(v,w))=w$, and its image is called $[[v,w]]$.

-   For any $v,w\in T$, the only non-self-intersecting path going from $v$ to $w$ is $[[v,w]]$, i.e. for any continuous injective function $s\mapsto
    v_{s}$ from $[0,1]$ to $T$ with $v_{0}=v$ and $v_{1}=w$, $\{v_{s}:s\in
    \lbrack 0,1]\}=[[v,w]]$.

We will furthermore consider $\mathbb{R}$-trees that are *rooted*, that is, one vertex is distinguished as being the root, and we call 