Correction terms for the height of weighted recursive trees

Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached parent chosen randomly proportionally its weight. Under some assumptions on the sequence of weights, first order for height such has been recently established one authors. In this paper, we obtain second and third orders in asymptotic expansion weighted trees, under similar assumptions. Our methods inspired from those used prove results branching random walks. also apply related model growing called preferential attachment tree additive fitnesses.