Kemp (1989, 1995, 1998) studied models of trees of trees with applications to computer science such as search trees for multidimensional keys. The present talk focuses on their properties when generated at random either uniformly or via reducible multi-type branching processes. Our main result is the discovery that contrarily to other combinatorial models their scaling limit is not the Brownian tree, but a novel inhomogeneous random tree whose construction is related to a two-parameter Poisson-Dirichlet process.