On the expected width function for topologically random channel networks

An idealized river-channel network is represented by a trivalent planted plane tree, the root of which corresponds to the outlet of the network. A link of the network is any segment between a source and a junction, two successive junctions, or the outlet and a junction. For any x≧0, the width of the network is the number of links with the property that the distance of the downstream junction from the outlet is ≦x, and the distance of the upstream junction to the outlet is > x. Expressions are obtained for the expected width conditioned on N, (N, M), and (N, D), where N is the magnitude, M the order, and D the diameter of the network, under the assumption that the network is drawn from an infinite topologically random population and the link lengths are random.