Limits of biconditioned Bienayme-Galton-Watson trees

We study the limiting behavior of a Bienayme-Galton-Watson tree conditioned to have a large number of vertices and either a fixed number of leaves or a fixed number of internal nodes. The first biconditioning gives a universal result with respect to the offspring distribution. In contrast, the second case leads to a variety of limiting behaviors, ranging from condensation phenomena to more elongated tree structures, depending on the properties of the offspring distribution. To prove these results, we use tools from conditioned random walk theory and from analytic combinatorics.