An (unrooted) d-ary tree is a tree in which every internal vertex has degree d+1. In this paper, we show for every fixed d≥2 that d-ary caterpillars have the minimum number of dominating sets among d-ary trees of a given order. We also determine the maximum number of dominating sets in binary trees (the special case d=2) and classify the extremal trees, which are also unique.