Pruning of CRT-sub-trees

We study the pruning process developed by Abraham and Delmas (2012) on the discrete Galton-Watson sub-trees of the L\'{e}vy tree which are obtained by considering the minimal sub-tree connecting the root and leaves chosen uniformly at rate $\lambda$, see Duquesne and Le Gall (2002). The tree-valued process, as $\lambda$ increases, has been studied by Duquesne and Winkel (2007). Notice that we have a tree-valued process indexed by two parameters the pruning parameter $\theta$ and the intensity $\lambda$. Our main results are: construction and marginals of the pruning process, representation of the pruning process (forward in time that is as $\theta$ increases) and description of the growing process (backward in time that is as $\theta$ decreases) and distribution of the ascension time (or explosion time of the backward process) as well as the tree at the ascension time. A by-product of our result is that the super-critical L\'{e}vy trees independently introduced by Abraham and Delmas (2012) and Duquesne and Winkel (2007) coincide. This work is also related to the pruning of discrete Galton-Watson trees studied by Abraham, Delmas and He (2012).