Twenty years ago, Luczak and Winkler proved that a uniformly random plane d-ary tree with n nodes can be constructed from a uniformly random one with n-1 nodes, by adding leaves at a well-chosen place. I will discuss generalizations of this construction to other models of discrete random trees under log-concavity type assumptions, as well as the continuum counterpart of the problem. Along the way, we will see that the Aldous' Continuum Random Tree can be "grown by the leaves", and that the resulting dynamics is the scaling limit of the Luczak-Winkler process.