Empirical observations have established connections between river network geometry and various hydrophysical quantities of interest. Since rivers can be decomposed into basic components known as links, one would like to understand the physical processes at work in link formation and maintenance. The author develops a natural stochastic geometric model for this problem, for the particular type of link known as exterior links. In the model, the distribution of distance from a uniformly distributed point to a fixed graph is computed. This model yields an approximate expression for the distribution of length of exterior links that incorporates junction angles and drainage density, and compares favorably with observed length distributions. The author goes on to investigate related mathematical questions of independent interest, such as the case where the previously mentioned graph is itself a realization of a random process, and in so doing derives a formula for the first contact distribution of a general random Voronoi tesselation (also associated with the names of Dirichlet and Thiessen). Since this random tesselation is a natural starting point for modelling spatial processes in a wide variety of fields, these results should find immediate applications. It is also shown how these results can be interpreted as a generalization of a classical problem considered by Buffon. / Graduation date: 1991