A conjecture on different central parts of binary trees

TL;DR

This paper investigates the extremal properties of binary trees related to the distances between their central parts, confirming a conjecture that specific structured trees maximize these distances.

Contribution

The paper proves a conjecture about the maximum distances between central parts of binary trees and characterizes extremal trees for these measures.

Findings

Confirmed the conjecture regarding maximum distances between central parts.

Identified the binary trees that minimize root-containing subtrees.

Characterized the binary trees that maximize distances between central parts.

Abstract

Let $\Omega_n$ be the family of binary trees on $n$ vertices obtained by identifying the root of an rgood binary tree with a vertex of maximum eccentricity of a binary caterpillar. In the paper titled "On different middle parts of a tree (The electronic journal of combinatorics, 25 (2018), no. 3, paper 3.17, 32 pp)", Smith et al. conjectured that among all binary trees on $n$ vertices the pairwise distance between any two of center, centroid and subtree core is maximized by some member of the family $\Omega_n$. We first obtain the rooted binary tree which minimizes the number of root containing subtrees and then prove this conjecture. We also obtain the binary trees which maximize these distances.