We study the  pruning process developed by Abraham  and Delmas (2012) on
the discrete Galton-Watson sub-trees of the Lévy 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 Lévy 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).