It is well known that in the dynamics of a piecewise strictly monotone (that is, piecewise embedding) map $f$ on an interval, the topological entropy can be expressed in terms of the growth of the number (that is, the lap number) of strictly monotone intervals for $f^n$. Recently, there has been an increase in the importance of fractal sets in the sciences, and many geometric and dynamical properties of fractal sets have been studied. In the present paper, we shall study topological entropy of some maps on regular curves, which are contained in the class of fractal sets. We generalize the theorem of Misiurewicz–Szlenk and Young to the cases of regular curves and dendrites.