  We consider the fragmentation at nodes of the L\'{e}vy continuous random tree
introduced in a previous paper. In this framework we compute the asymptotic for
the number of small fragments at time $\theta$. This limit is increasing in
$\theta$ and discontinuous. In the $\alpha$-stable case the fragmentation is
self-similar with index $1/\alpha$, with $\alpha \in (1,2)$ and the results are
close to those Bertoin obtained for general self-similar fragmentations but
with an additional assumtion which is not fulfilled here.
