The forest associated with the record process on a L\'evy tree

We perform a pruning procedure on a L\'evy tree and instead of throwing away the removed sub-tree, we regraft it on a given branch (not related to the L\'evy tree). We prove that the tree constructed by regrafting is distributed as the original L\'evy tree, generalizing a result where only Aldous's tree is considered. As a consequence, we obtain that the quantity which represents in some sense the number of cuts needed to isolate the root of the tree, is distributed as the height of a leaf picked at random in the L\'evy tree.