STREAM networks evolve by headward growth and branching away from escarpments such as rift margins. The structure of these networks and their topographic relief are known to be fractal1–3, but no model so far has been able to generate the observed scaling properties. Here I present a statistical model of network growth in which stream heads branch and propagate at a rate that depends only on the local strength of the substrate. This model corresponds to the process of invasion percolation4, with the added requirement of self-avoidance; it is a self-organized critical system5 with properties similar to those of standard percolation models6. A description based on self-avoiding invasion percolation reproduces the known scaling behaviour of stream networks, and may provide a valuable tool for delineation of drainage patterns from digital topographic data sets7,8.