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

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... mula for the joint law of the height and diameter of the Brownian tree, which is a new result. Supersedes arXiv:1503.05014v1. 2 , converges in distribution to the Brownian tree (also called Continuum Random Tree) that is a random compact metric space. From this, Aldous has deduced that ∆ has the same distribution as the diameter of the Brownian tree: see [3], Section 3.4, (though formula (41) there is not accurate). As proved by Aldous [4] and by Le Gall [18], the Brownian tree is coded by the normalized Brownian excursion of length 1 (see below for more detail). Then, the question was raised by Aldous [3] that whether we can establish (1.2) directly from computations