  We consider a fragmentation of discrete trees where the internal vertices are
deleted independently at a rate proportional to their degree. Informally, the
associated cut-tree represents the genealogy of the nested connected components
created by this process. We essentially work in the setting of Galton-Watson
trees with offspring distribution belonging to the domain of attraction of a
stable law of index $\alpha\in(1,2)$. Our main result is that, for a sequence
of such trees $\mathcal{T}_n$ conditioned to have size $n$, the corresponding
rescaled cut-trees converge in distribution to the stable tree of index
$\alpha$, in the sense induced by the Gromov-Prokhorov topology. This gives an
analogue of a result obtained by Bertoin and Miermont in the case of
Galton-Watson trees with finite variance.

Probability