The progress of the successive rounds of catalytic conversion of substrates into product(s) by a single enzyme is characterized by the distribution of turnover times. Establishing the most general form of dependence of this distribution on the substrate concentration [S] is one of the fundamental challenges in single molecule enzymology. The distribution of the times of dwell of a molecular motor at the successive positions on its track is an analogous quantity. We derive approximate series expansions for the [ATP]-dependence of the first two moments of the dwell time distributions of motors that catalyze hydrolysis of ATP to draw input energy. Comparison between our results for motors with branched pathways and the corresponding expressions reported earlier for linear enzymatic pathways provides deep insight into the effects of the branches. Such insight is likely to help in discovering the most general form of [S]-dependence of these fundamental distributions.