  Weighted recursive trees are built by adding successively vertices with
predetermined weights to a tree: each new vertex is attached to a parent chosen
randomly proportionally to its weight. Under some assumptions on the sequence
of weights, the first order for the height of such trees has been recently
established in [Electron. J. Probab. 26 (2021), Paper No. 80] by one of the
authors. In this paper, we obtain the second and third orders in the asymptotic
expansion of the height of weighted recursive trees, under similar assumptions.
Our methods are inspired from those used to prove similar results for branching
random walks. Our results also apply to a related model of growing trees,
called the preferential attachment tree with additive fitnesses.
