The tight upper bound on the state complexity of the reverse of R-trivial and J -trivial regular languages of the state complexity n is 2 n−1 . The witness is ternary for R-trivial regular languages and (n − 1)ary for J -trivial regular languages. In this paper, we prove that the bound can be met neither by a binary R-trivial regular language nor by a J -trivial regular language over an (n − 2)-element alphabet. We provide a characterization of tight bounds for R-trivial regular languages depending on the state complexity of the language and the size of its alphabet. We show the tight bound for J -trivial regular languages over an (n − 2)-element alphabet and a few tight bounds for binary J -trivial regular languages. The case of J -trivial regular languages over an (n−k)element alphabet, for 2 ≤ k ≤ n − 3, is open.