fragmentation occurs, the largest resulting fragment is not too small. In particular, it is always satisfied in the case of fragmentations for which $\nu
(s_{N+1}>0)=0$, since then $s_{1}>1/N$ for $\nu $-a.e. $\mathbf{s}$. Yet, we point out that when $\int_{S}\left( s_{1}^{-1}-1\right) \nu (\mathrm{d}\mathbf{s})=\infty $, one anyway obtains the following bounds for the Hausdorff dimension of $\mathcal{L}(\mathcal{T}_{F})$: $$\frac{\varrho }{\left| \alpha \right| }\leq \dim_{\cal H}
\left( \mathcal{L}(\mathcal{T}_{F})\right) \leq \frac{1}{\left| \alpha
\right| }\text{ a.s}.$$ where $$\varrho :=\sup \left\{ p\leq 1:\int_{S}\left( s_{1}^{-p}-1\right) \nu ({\mathrm{d}\mathbf{s}})<\infty \right\} .  \label{219}$$

It is worth noting that these results allow as a special case to compute the Hausdorff dimension of the so-called *stable trees* of Duquesne and Le Gall [@duqleg02], which were used to construct fragmentations in the manner of Theorem \[T1\] in [@mierfmoins]. The dimension of the stable tree (as well as finer results of Hausdorff measures on more general Lévy trees) has been obtained independently in [@duqlegprep]. The stable tree is a CRT whose law depends on parameter $\beta\in(1,2]$, and it satisfies the required self-similarity property of Proposition \[reciproque\] with index $1/\beta-1$. We check that the associated dislocation measure satisfies the integrability condition of Theorem \[T2\] in Sect. \[checkst\], so that

\[stable\] Fix $\beta \in (1,2]$. The $\beta 