Asymptotic results on Hoppe trees and its variations

A uniform recursive tree on $n$ vertices is a random tree where each possible $(n-1)!$ labeled recursive rooted tree is selected with equal probability. In this paper we introduce and study weighted trees, a non-uniform recursive tree model departing from the recently introduced Hoppe trees. This class generalizes both uniform recursive trees and Hoppe trees. The generalization provides diversity among the nodes, making the model more flexible for applications. We also analyze the number of leaves, the height, the depth, the number of branches, and the size of the largest branch in these weighted trees.