Asymptotic results on Hoppe trees and its variations

TL;DR

This paper introduces weighted recursive trees, generalizing uniform and Hoppe trees, and analyzes key structural properties to enhance their applicability in various contexts.

Contribution

It presents a new weighted tree model that extends Hoppe trees and studies its structural characteristics and potential applications.

Findings

Weighted trees generalize uniform and Hoppe trees.

Analysis of leaves, height, depth, branches, and largest branch.

Provides insights into the diversity and flexibility of recursive tree models.

Abstract

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.