We study the surface growth of linear filaments into networks where filament orientation is determined by the substrate lattice and growth stops when a filament encounters an existing filament. Depending on the orientational symmetry of the substrate lattice, triaxial, biaxial, rectangular, and isotropic networks can form. By using mean-field theory and simulation, we show that the filament length distribution within the network crosses over from exponential for long filaments, to a power-law for shorter ones, with the power law exponent in the 2.5-3.0 range. We also discuss a Computer-Aided Feature Extraction (CAFE) program that can identify individual filaments in experimental images and allows us to compare simulation and experiment directly. Because the morphology of the network reflects its growth history, an analysis of experimental images by CAFE provides kinetic information about the network assembly process.