A Conjecture on Different Central Parts of Binary Trees

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.