A cellular automaton model proposed by Motoike includes reaction-diffusion dynamics in a simple manner (Motoike, J. Phys. Soc. Jpn. 2007). The semi-random grid adopted in this model is designed to describe biological phenomena that involve intrinsic fluctuations. It has been shown that this model can exhibit branching patterns reminiscent of neural dendrites, whose formation may be regulated by excitation signals as suggested by experimental results. From a computational viewpoint, a random grid can be regarded as representing intrinsic spatial noise. In this study, we firstly compared the features of the patterns obtained from numerical simulations using regular and semi-random square grids. It was demonstrated that the directions of the path growth tended to be orthogonal or parallel to the grid owing to the anisotropy of the regular grid. We found that, as the parameter values are varied, the numbers of endpoints change continuously for patterns exhibited by a semi-random grid, whereas, they change discontinuously for a regular grid. Next, we investigated the patterns of path formation when excitation signals were added at spatially random points. The patterns obtained under these conditions exhibited highly complex and random shapes of branches. It was also revealed that the patterns in the case of random inputs have endpoints that are more numerous than in the case of a spatially fixed input. This result suggested that the patterns represent the history of the spatial history of excitation signal inputs.