a positive fraction of the blocks of $\Pi(t)$ are singletons for some $t>0$. This last property of self-similar fragmentations with negative index allows to build a collection of trees with edge-lengths.

Trees with edge-lengths {#twel}
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A tree is a finite connected graph with no cycles. It is *rooted* when a particular vertex (the root) is distinguished from the others, in this case the edges are by convention oriented, pointing from the root, and we define the out-degree of a vertex $v$ as being the number of edges that point outward $v$. A *leaf* in a rooted tree is a vertex with out-degree $0$. For $k\geq 1$, let $\mathbf{T}_k$ be the set of rooted trees with exactly $k$ labeled leaves (the names of the labels may change according to what we see fit), the other vertices (except the root) begin unlabeled , and such that the root is the only vertex that has out-degree $1$. If $\mathbf{t}\in \mathbf{T}_k$, we let $E(\mathbf{t})$ be the set of its edges.

A *tree with edge-lengths* is a pair $\vartheta =(\mathbf{t},\mathbf{e}) $ for $\mathbf{t}\in \bigcup_{k\geq 1}\mathbf{T}_{k}$ and $\mathbf{e}=(e_{i},i\in E(t))\in (\mathbb{R}_{+}\setminus \{0\})^{E(\mathbf{t})}$. Call $\mathbf{t}$ the *skeleton* of $\vartheta $. Such a tree is naturally equipped with 