We investigate optimal predictors of the peak (S) and distance to peak (T) of the width function of drainage networks under the assumption that the networks are topologically random with independent and exponentially distributed link lengths. Analytical results are derived using the fact that, under these assumptions, the width function is a homogeneous Markov birth-death process. In particular, exact expressions are derived for the asymptotic conditional expectations of S and T given network magnitude N and given mainstream length H. In addition, a simulation study is performed to examine various predictors of S and T, including N, H, and basin morphometric properties; non-asymptotic conditional expectations and variances are estimated. The best single predictor of S is N, of T is H, and of the scaled peak (S divided by the area under the width function) is H. Finally, expressions tested on a set of drainage basins from the state of Wyoming perform reasonably well in predicting S and T despite probable violations of the original assumptions. ?? 1989 Springer-Verlag.