leaves must be uncountable and have no isolated point. Also, the definition of a CRT is a little inaccurate as we did not endow the space of ${\mathbb{R}}$-trees with a $\sigma$-field. This problem is in fact circumvented by the fact that CRTs are in fact entirely described by the sequence of their [*marginals*]{}, that is, of the subtrees spanned by the root and $k$ leaves chosen with law $\mu$ given $\mu$, and these subtrees, which are interpreted as finite [*trees with edge-lengths*]{}, are random variables (see Sect. \[twel\]). The reader should keep in mind that by the “law” of a CRT we mean the sequence of these marginals.

For $(T,\mu )$ a continuum tree, and for every $t\geq 0$, let $T_{1}(t),T_{2}(t),\ldots $ be the tree components of $\{v\in T:\mathrm{ht}
(v)>t\}$, ranked by decreasing order of $\mu $-mass. A continuum random tree $(T,\mu )$ is said to be *self-similar* with index $\alpha <0$ if for every $t\geq 0$, conditionally on $(\mu (T_{i}(t)),i\geq 1)$, $
(T_{i}(t),i\geq 0)$ has the same law as $(\mu (T_{i}(t))^{-\alpha }\otimes
T^{(i)},i\geq 1)$ where the $T^{(i)}$’s are independent copies of $T$.

Our first result is

\[T1\] Let $F$ be a ranked self-similar fragmentation process with characteristic $3$-tuple $(\alpha 