RANDOM TREES CONSTRUCTED BY AGGREGATION

We study a general procedure that builds random R-trees by gluing recursively a new branch on a uniform point of the pre-existing tree. The aim of this paper is to sec how the asymptotic behavior of the sequence of lengths of branches influences some geometric properties of the limiting tree, such as compactness and Hausdorff dimension. In particular, when the sequence of lengths of branches behaves roughly like n(-alpha) for some alpha is an element of (0, 1]. we show that the limiting tree is a compact random tree of Hausdorff dimension alpha(-1). This encompasses the famous construction of the Brownian tree of Aldous. When alpha > 1, the limiting tree is thinner and its IIausdorff dimension is always 1. In that case, we show that alpha(-1) corresponds to the dimension of the set of leaves of the tree.