We investigate the connection between surface elevation and the growth and scaling of river networks. Three planar models (Scheidegger, Eden, and invasion percolation) are first considered. These models develop aggregating networks according to stochastic rules but do not simulate erosion because the network growth is independent of the surface elevation. We show that none of these planar growth models produces scaling results consistent with observations for natural river basins. We then modify the models to include elevation, simulating the effects of fluvial erosion by enforcing the slope-area relationship. The resulting configurations have scaling properties that still depend on the model (Scheidegger, Eden, or invasion percolation) but are closer to natural river networks when compared with those from the planar growth rules. We conclude that inclusion of the vertical dimension in these three models is critical for explaining the formation and regularities of fluvial networks. (C) 2001 Elsevier Science BN. All rights reserved.