  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.
