In nonequilibrium growth such as diffusion-limited aggregation (DLA), the growth-site probability distribution characterizes these growth processes. By solving the Laplace equation numerically, we calculate the growth probability Pg(x) at the perimeter site x of clusters for the DLA and its generalized version called the model, and obtain the generalized dimension D(q) and the f spectrum proposed by Halsey et al. [Phys. Rev. A 33, 1141 (1986)]. It is found that D(q) depends strongly on q and that the f spectrum is continuous. Our results suggest that these growth processes cannot be described by a simple scaling theory with a few scaling exponents. This is in clear contrast to the Botet-Jullien model [Phys. Rev. Lett. 55, 1943 (1985)] which yields equilibrium patterns whose D(q) is constant. It is also found that the information dimension D(1) which represents the properties of the unscreened surface is in good agreement with our recent theory.