Height and diameter of Brownian tree

By computations on generating functions, Szekeres proved in 1983 that the law of the diameter of a uniformly distributed rooted labelled tree with n vertices, rescaled by a factor n^{-1/2}, converges to a distribution whose density is explicit. Aldous observed in 1991 that this limiting distribution is the law of the diameter of the Brownian tree. In our article, we provide a computation of this law which is directly based on the normalized Brownian excursion. Moreover, we provide an explicit formula for the joint law of the height and diameter of the Brownian tree, which is a new result.