We perform a pruning procedure on a Lévy tree and instead of throwing away the removed
sub-tree, we regraft it on a given branch (not related to the Lévy tree). We prove that the tree
constructed by regrafting is distributed as the original Lévy tree, generalizing a result of
Addario-Berry, Broutin and Holmgren where only Aldous's tree is considered. As a
consequence, we obtain that the “average pruning time” of a leaf is distributed as the height
of a leaf picked at random in the Lévy tree.