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 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 Lévy tree.