ranked self-similar fragmentation with characteristics $(\alpha ,c,\nu )$ satisfying the hypotheses of Theorem \[T1\]. Writing $\dim_{\cal H}$ for Hausdorff dimension, one has $$\dim_{\cal H}\left(\mathcal{L}(\mathcal{T}_{F})\right) =\frac{1}{\left| \alpha \right| }\text{ a.s}.$$ as soon as $\int_{S}\left( s_{1}^{-1}-1\right) \nu ({\mathrm{d}\mathbf{s}})<\infty $

Some comments about this formula. First, notice that under the extra integrability assumption on $\nu $, the dimension of the whole tree is $\dim_{\cal H}(\mathcal{T}_{F})=(1/|\alpha |)\vee 1$ because the skeleton $\mathcal{S}(\mathcal{T}_{F})$ has dimension $1$ as a countable union of segments. The value $-1$ is therefore critical for $\alpha $, since the above formula shows that the dimension of $\mathcal{T}_{F}$ as to be $1$ as soon as $\alpha \leq -1$. It was shown in a previous work by Bertoin [@bertafrag02] that when $\alpha <-1$, for every fixed $t$ the number of fragments at time $t$ is a.s. finite, so that $-1$ is indeed the threshold under which fragments decay extremely fast. One should then picture the CRT $\mathcal{T}_{F}$ as a “dead tree” looking like a handful of thin sticks connected to each other, while when $|\alpha |<1$ the tree looks more like a dense “bush”. Last, the integrability assumption in the theorem seems to be reasonably mild; its heuristic meaning is that when a 