  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).
