Theorem \[T1\]. Then $\dim_{\cal H}\left(
\mathrm{d}M_{F}\right) \leq 1\wedge 1/\left| \alpha \right| $ a.s.

Notice that this result re-implies the fact from [@haas03] that the height profile does not exist as soon as $|\alpha |\geq 1$.

The last motivation of this paper (Sect. \[heightprocess\]) is about relations between CRTs and their so-called encoding *height processes*. The fragmentation $F_B$ of [@bertsfrag02], as well as the fragmentations from [@mierfmoins], were defined out of certain random functions $(H_s,0\leq s\leq 1)$. Let us describe briefly the construction of $F_B$. Let $B^{\mathrm{exc}}$ be the standard Brownian excursion with duration $1$, and consider the open set $\{s\in [0,1]:2B^{\mathrm{exc}}_s>t\}$. Write $F(t)$ for the decreasing sequence of the lengths of its interval components. Then $F$ has the same law as the fragmentation $F_B^{\prime}$ defined out of the Brownian CRT in the same way as in Theorem \[T1\]. This is immediate from the description of Le Gall [@legall93] and Aldous [@aldouscrt93] of the Brownian tree as being encoded in the Brownian excursion. To be concise, define a pseudo-metric on $[0,1]$ by letting $\overline{d}(s,s^{\prime})=
2B^{\mathrm{exc}}_s+2B^{\mathrm{exc}}_{s^{\prime}}-4\inf_{u\in
[s,s^{\prime}]}B^{\mathrm{exc}}_u$, with the convention that $[s,s^{\prime}]=[s^{\prime},s]$ if $s^{\prime}<s$. We can define a true metric space by taking the quotient with respect to the equivalence relation $s\equiv 