The self-similar tree model specifies a general procedure which can be used to define the concept of a fractal tree graph. This procedure produces a fractal tree from an infinite sequence of nonnegative integers, called the generators of the tree. Many different tree graph models have been proposed as models for the branching or topological structure of river networks, in an effort to explain a number of different scaling laws that have been observed empirically in studies of river systems. Most of these models can be seen to be special cases of the self-similar tree model. This tree model satisfies several recursive formulae, the asymptotic behavior of which can be shown, using generating functions, to produce scaling laws like those observed. This model has started to see applications to other branching systems, such as the root systems of plants. I will also give an example of a closed-form solution to Laplace's equation that has branching structure somewhat similar to what is observed for systems of river valleys. This solution may have application to understanding the dendritic patterns that are observed in dielectric breakdown