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 -vertex non-caterpillars with given diameter , we find the unique tree formed by attaching the path and pendant vertices to a center of the path on vertices with minimum Wiener index, where , and we determine the -vertex non-caterpillars with the th greatest reverse Wiener indices for all up to if and up to or depending on whether is an integer or not for even , and or depending on whether is an integer or not for odd if .