Asymptotics of heights in random trees constructed by aggregation

To each sequence (a(n)) of positive real numbers we associate a growing sequence (T-n) of continuous trees built recursively by gluing at step n a segment of length a(n) on a uniform point of the pre-existing tree, starting from a segment T-1 of length a(1). Previous works [ 5, 10] on that model focus on the influence of (a(n)) on the compactness and Hausdorff dimension of the limiting tree. Here we consider the cases where the sequence (a(n)) is regularly varying with a non-negative index, so that the sequence (T-n) explodes. We determine the asymptotics of the height of T-n and of the subtrees of T-n spanned by the root and l points picked uniformly at random and independently in T-n, for all l is an element of IN.