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 n -vertex non-caterpillars with given diameter d , we find the unique tree formed by attaching the path P 2 and n − d − 3 pendant vertices to a center of the path on d + 1 vertices with minimum Wiener index, where 4 ≤ d ≤ n − 3 , and we determine the n -vertex non-caterpillars with the k th greatest reverse Wiener indices for all k up to ⌊ n − 3 2 ⌋ if 8 ≤ n ≤ 26 and up to ⌊ n − 3 2 ⌋ − ⌈ a ( n ) ⌉ + 1 or ⌊ n − 3 2 ⌋ − ⌈ a ( n ) ⌉ + 2 depending on whether a ( n ) = n − 3 2 − n 2 + 2 n + 2 8 is an integer or not for even n , and ⌊ n − 3 2 ⌋ − ⌈ b ( n ) ⌉ + 1 or ⌊ n − 3 2 ⌋ − ⌈ b ( n ) ⌉ + 2 depending on whether b ( n ) = n − 3 2 − n 2 + 2 n + 5 8 is an integer or not for odd n if n ≥ 27 .