s^{\prime}\iff \overline{d}(s,s^{\prime})=0$. Call $( \mathcal{T }_B,d)$ this metric space. Write $\mu_B$ for the measure induced on $\mathcal{T }_B$ by Lebesgue measure on $[0,1]$. Then $( \mathcal{T }_B,\mu_B) $ is the Brownian CRT, and the equality in law of the fragmentations $F_B$ and $F_B^{\prime}$ follows immediately from the definition of the mass measure. Our next result generalizes this construction.

\[T3\] Let $F$ be a ranked self-similar fragmentation with same hypotheses as in Theorem \[T1\], and suppose $\nu $ has infinite total mass. Then there exists a continuous random function $(H_{F}(s),0\leq s\leq
1)$, called the height function, such that $H_{F}(0)=H_{F}(1)$, $H_{F}(s)>0$ for every $s\in (0,1)$, and such that $F$ has the same law as the fragmentation $F^{\prime }$ defined by: $F^{\prime }(t)$ is the decreasing rearrangement of the lengths of the interval components of the open set $I_{F}(t)=\{s\in (0,1):H_{F}(s)>t\}$.

An interesting point in this construction is also that it shows that a large class of self-similar fragmentation with negative index has a natural *interval representation*, given by $(I_F(t),t\geq 0)$.

In parallel to the computation of the Hausdorff dimension of the CRTs built above, we are able to estimate Hölder coefficients for the height processes of these CRTs. Our result is

\[T4\] Suppose 