  We study a general procedure that builds random $\mathbb R$-trees by gluing
recursively a new branch on a uniform point of the pre-existing tree. The aim
of this paper is to see 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 $\alpha \in
(0,1]$, 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
Hausdorff dimension is always 1. In that case, we show that $ \alpha^{-1}$
corresponds to the dimension of the set of leaves of the tree.
