In order to investigate the effect of the structure of a non-complete substrate on the dynamic behaviors of a growing surface, the restricted solid-on-solid model on Sierpinski arrowhead and Crab fractal substrates, which have the same fractal dimensions but of different spectrum dimensions, are extensively studied by means of numerical simulations. The surface width and the maximal height of the saturated surface are calculated. It is found that the microscopic structure of the substrates affects significantly the dynamic properties of the surfaces. Although the restricted solid-on-solid model evolving on two kinds of fractal substrates exhibits dynamic scaling behavior, the standard Family-Vicsek scaling is still satisfied for different dynamic scaling exponents. The maximal height of the width of saturated surface can be fitted by Asym2Sig distribution, not by the three kinds of usual extreme statistical distribution, i.e. Weibull, Gumbel, and Frechet distributions.