  Aldous, Evans and Pitman (1998) studied the behavior of the fragmentation
process derived from deleting the edges of a uniform random tree on $n$
labelled vertices. In particular, they showed that, after proper rescaling, the
above fragmentation process converges as $n \rightarrow \infty$ to the
fragmentation process of the Brownian CRT obtained by cutting-down the Brownian
CRT along its skeleton in a Poisson manner.
  In this work, we continue the above investigation and study the fragmentation
process obtained by deleting randomly chosen edges from a critical
Galton-Watson tree $\mathbf{t}_{n}$ conditioned on having $n$ vertices, whose
offspring distribution belongs to the domain of attraction of a stable law of
index $\alpha \in (1,2]$. Our main results establish that, after rescaling, the
fragmentation process of $\mathbf{t}_{n}$ converges as $n \rightarrow \infty$
to the fragmentation process obtained by cutting-down proportional to the
length on the skeleton of an $\alpha$-stable L\'evy tree of index $\alpha \in
(1,2]$. We further show that the latter can be constructed by considering the
partitions of the unit interval induced by the normalized $\alpha$-stable
L\'evy excursion with a deterministic drift studied by Miermont (2001). This
extends the result of Bertoin (2000) on the fragmentation process of the
Brownian CRT.
