  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 [arXiv:1904.07115] 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.
