Distance-hereditary graphs form an important class of graphs, from the theoretical point of view, due to the fact that they are the totally decomposable graphs for the split-decomposition. The previous best enumerative result for these graphs is from Nakano et al. (J. Comp. Sci. Tech., 2007), who have proven that the number of distance-hereditary graphs on n vertices is bounded by 2[3.59n]. In this paper, using classical tools of enumerative combinatorics, we improve on this result by providing an exact enumeration and full asymptotic of distance-hereditary graphs, which allows to show that the number of distance-hereditary graphs on n vertices is tightly bounded by (7.24975⋯)n - opening the perspective such graphs could be encoded on 3n bits. We also provide the exact enumeration and full asymptoticss of an important subclass, the 3-leaf power graphs. Our work illustrates the power of revisiting graph decomposition results through the framework of analytic combinatorics.