  We study a natural fragmentation process of the so-called stable tree
introduced by Duquesne and Le Gall, which consists in removing the nodes of the
tree according to a certain procedure that makes the fragmentation self-similar
with positive index. Explicit formulas for the semigroup are given, and we
provide asymptotic results. We also give an alternative construction of this
fragmentation, using paths of Levy processes, hence echoing the two alternative
constructions of the standard additive coalescent by fragmenting the Brownian
continuum random tree or using Brownian paths, respectively due to
Aldous-Pitman and Bertoin.
