A Cantor-Bendixson Rank for Siblings of Trees

TL;DR

This paper introduces a new way to measure the complexity of trees using a Cantor-Bendixson rank, and provides a novel representation of trees to analyze their siblings, advancing understanding in this area.

Contribution

It defines the Cantor-Bendixson rank for trees and presents a new tree representation to study the number of sibling trees, addressing a conjecture in the field.

Findings

Developed a Cantor-Bendixson rank for trees

Represented trees as leafless trees with attached leafy trees

Obtained partial results on the sibling count conjecture

Abstract

Similar to topological spaces, we introduce the Cantor-Bendixson rank of a tree $T$ by repeatedly removing the leaves and the isolated vertices of $T$ using transfinite recursion. Then, we give a representation of a tree $T$ as a leafless tree $T^\infty$ with some leafy trees attached to $T^\infty$. With this representation at our disposal, we count the siblings of a tree and obtain partial results towards a conjecture of Bonato and Tardif.