Natural river channel networks have been shown in empirical studies to exhibit power-law scaling behavior characteristic of self-similar and self-affine structures. Of particular interest is to describe how the distribution of distance to the outlet changes as a function of network size. In this paper, networks are modeled as random self-similar rooted tree graphs and scaling of distance to the root is studied using methods in stochastic branching theory. In particular, the asymptotic expectation of the width function (number of nodes as a function of distance to the outlet) is derived under conditions on the replacement generators. It is demonstrated further that the branching number describing rate of growth of node distance to the outlet is identical to the length ratio under a Horton-Strahler ordering scheme as order gets large, again under certain restrictions on the generators. These results are discussed in relation to drainage basin allometry and an application to an actual drainage network is presented. ?? World Scientific Publishing Company.