A model for drainage networks and their associated landscapes has been developed. This model is based on the minimum energy dissipation principle and an empirical relationship s~Q&agr; between the slope s of each link in a channel network and the mean annual discharge Q that flows through it. The model was studied using six different values for the exponent &agr;. The surfaces of these minimum energy dissipation drainage networks appear to be more complex than simple self-affine fractals. The drainage basins in these optimal drainage networks have power law size (area) distributions with a universal exponent, which is independent of the value of &agr;. For the minimum energy dissipation drainage networks obtained using a particular value for &agr;, the basin shapes are similar to each other. An empirical relationship has also been found to relate the basins' ''length-width'' ratio to the value of &agr;.