Tree evolution processes for bucket increasing trees

TL;DR

This paper characterizes all bucket increasing trees generated by tree evolution processes, extending previous results for ordinary increasing trees and providing new insights into their structural properties.

Contribution

It offers a complete characterization of bucket increasing trees via tree evolution processes and extends existing results to include second order properties.

Findings

Complete characterization of bucket increasing trees

Equivalent properties for these trees

Second order results for descendants of a label

Abstract

We provide a fundamental result for bucket increasing trees, which gives a complete characterization of all families of bucket increasing trees that can be generated by a tree evolution process. We also provide several equivalent properties, complementing and extending earlier results for ordinary increasing trees to bucket trees. Additionally, we state second order results for the number of descendants of label $j$, again extending earlier results in the literature.