Abstract: We examine a certain class of three-regular graphs used to represent one-face maps. Numerical studies have revealed that the eigenvalue statistics of these graphs are the same as those of much larger and more widely studied classes of random matrices. If we limit our studies to three-regular graphs representing genus zero one-face maps, the eigenvalue statistics are strikingly different. The embedded graph of a genus zero one-face map is a tree, and there is a correlation between its vertices and the primitive cycles of the associated three-regular graph. In this talk, we examine the structure and eigenvalue statistics of these embedded planar trees. In particular, we show how the Dyck path representation can be used to recast questions about the probabilistic structure of random planar trees into straightforward counting problems. Using this Dyck path approach, we find: 1. the expected number of degree k vertices adjacent to j degree d vertices in a random planar tree, 2. the structure of the planar tree's adjacency matrix under a natural labeling of the vertices, and 3. an explanation for the existence of eigenvalues with multiplicity greater than one in the tree's spectrum.