On ordinary and reverse Wiener indices of non-caterpillars

A tree is known as a caterpillar if the removal of all pendant vertices makes it a path. Otherwise, it is a non-caterpillar. From among all nn-vertex non-caterpillars with given diameter dd, we find the unique tree formed by attaching the path P2P2 and n−d−3n−d−3 pendant vertices to a center of the path on d+1d+1 vertices with minimum Wiener index, where 4≤d≤n−34≤d≤n−3, and we determine the nn-vertex non-caterpillars with the kkth greatest reverse Wiener indices for all kk up to ⌊n−32⌋ if 8≤n≤268≤n≤26 and up to ⌊n−32⌋−⌈a(n)⌉+1 or ⌊n−32⌋−⌈a(n)⌉+2 depending on whether a(n)=n−32−n2+2n+28 is an integer or not for even nn, and ⌊n−32⌋−⌈b(n)⌉+1 or ⌊n−32⌋−⌈b(n)⌉+2 depending on whether b(n)=n−32−n2+2n+58 is an integer or not for odd nn if n≥27n≥27.