,c,\nu )$, with $\alpha <0$. Suppose also that $F$ is not constant, that $c=0$ and $\nu (\sum_{i}s_{i}<1)=0$. Then there exists an $\alpha $-self-similar CRT $(\mathcal{T}_{F},\mu _{F})$ such that, writing $F^{\prime }(t)$ for the decreasing sequence of masses of connected components of the open set $\{v\in \mathcal{T}_{F}:\mathrm{ht}(v)>t\}$, the process $(F^{\prime }(t),t\geq 0)$ has the same law as $F$. The tree $\mathcal{T}_{F}$ is leaf-dense if and only if $\nu $ has infinite total mass.

The next statement is a kind of converse to this theorem.

\[reciproque\] Let $(\mathcal{T},\mu )$ be a self-similar CRT with index $\alpha <0$. Then the process $F(t)=((\mu (\mathcal{T}_{i}(t),i\geq 1),t\geq
0)$ is a ranked self-similar fragmentation with index $\alpha $, it has no erosion and its dislocation measure $\nu $ satisfies $\nu
(\sum_{i}s_{i}<1)=0 $. Moreover, $\mathcal{T}_{F}$ and $\mathcal{T}$ have the same law.

These results are proved in Sect. \[crtf\]. There probably exists some notion of continuum random tree extending the former which would include fragmentations with erosion or with sudden loss of mass, but such fragmentations usually are less interesting.

The next result, to be proved in Sect. \[hausdorff\], deals with the Hausdorff dimension of the set of leaves of the CRT $\mathcal{T }_{F}$.

\[T2\] Let $F$ be a 