$-stable tree has Hausdorff dimension $\beta /(\beta -1)$.

An interesting process associated to a given continuum tree $(T,\mu)$ is the so-called *cumulative height profile* $\bar{W}_{T}(h)=\mu\{v\in T:\mathrm{ht}(v)\leq h\}$, which is non-decreasing and bounded by $1$ on $\mathbb{R}_+$. It may happen that the Stieltjes measure $\mathrm{d}\bar{W}_{T}(h)$ is absolutely continuous with respect to Lebesgue measure, in which case its density $(W_{T}(h),h\geq 0)$ is called the *height profile*, or *width process* of the tree. In our setting, for any fragmentation $F $ satisfying the hypotheses of Theorem \[T1\], the cumulative height profile has the following interpretation: one has $(\bar{W}_{\mathcal{T}_{F}}(h),h\geq 0)$ has the same law as $(M_F(h),h\geq 0)$, where $M_F(h)=1-\sum_{i\geq 1}F_i(h)$ is the total mass lost by the fragmentation at time $h$. Detailed conditions for existence (or non-existence) of the width profile $\mathrm{d} M_F(h)/\mathrm{d} h$ have been given in [@haas03]. It was also proved there that under some mild assumptions $\mathrm{dim\,} _{\mathcal{H}}\left( \mathrm{d} M_{F}\right) \geq 1\wedge A/\left|
\alpha \right| $ a.s., where $A$ is a $\nu $-dependent parameter introduced in $\left( \ref{278}\right) $, Section 3 below. The upper bound we obtain for $\dim_{\cal H}\left( \mathcal{L}(\mathcal{T}_{F})\right)
$ allows us to complete this result:

\[coromass\] Let $F$ be a ranked self-similar fragmentation with same hypotheses as in 