We consider the height process of a Levy process with no negative jumps, and its associated continuous tree representation. Using Levy snake tools developed by Duquesne and Le Gall, we construct a fragmentation process (at node and at height), which in the stable case corresponds to the self-similar fragmentation (at node and at height) described by Miermont. For the general fragmentation process we compute a family of dislocation measures. We compute also the asymptotic for the number of small fragments at time t. In the case of the fragmentation at node, this limit is increasing in t and discontinuous. In the alpha-stable case the fragmentation is self-similar and the results are close (but still different) to those obtained by Bertoin for general self-similar fragmentations under a strong additional assumtion which is not fulfilled here.