is infinite, into a height function. The set of leaves of the CRT then has a Hausdorff dimension  smaller than $1/\left| b\right| $ and the height function is $\gamma $-Hölder continuous for every $\gamma <\vartheta _{\text{low}}\wedge \left| b\right| .$ If moreover $\limsup_{x\rightarrow
0}x^{-a}\varsigma (x)<\infty $ for some $a<0$ and $\int_{S}\left(
s_{1}^{-1}-1\right) \nu (\mathrm{d}\mathbf{s})<\infty ,$ the Hausdorff dimension is larger than $1/\left| a\right| $ and the height function cannot have a Hölder coefficient $\gamma >\vartheta _{\text{sup}}\wedge \left|
a\right| $.

The CRT $\mathcal{T }_F$ {#crtf}
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Building the CRT $\mathcal{T }_F$ associated to a ranked fragmentation $F$ will be done by determining its “marginals”, i.e. the subtrees spanned by a finite but arbitrary number of randomly chosen leaves. To this purpose, it will be useful to use partitions-valued fragmentations, which we first define, as well as a certain family of trees with edge-lengths.

Exchangeable partitions and partitions-valued self-similar fragmentations {#pvssf}
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Let $\mathcal{P}_{\infty }$ be the set of (unordered) partitions of $\mathbb{N}=\{1,2,\ldots \}$ and $[n]=\{1,2,\ldots ,n\}$. We adopt the following ordering convention: for $\pi \in \mathcal{P}_{\infty }$, we let $(\pi
_{1},\pi _{2},\ldots )$ be the blocks of $\pi $, so that $\pi _{i}$ is the block containing $i$ provided that $i$ is 