Nelson, Garrett2025-04-152025-04-152025https://hdl.handle.net/2097/44929The Catalan numbers and mathematical objects enumerated by them have been studied since the 1700’s. The study of these objects, related objects, and bijections between them is called Catalan combinatorics. In this thesis, we discuss three areas of Catalan combinatorics. First, we discuss rational parking functions and the Pak-Stanley bijection from rational parking functions to alcoves of the Sommers region. Second, we discuss the combinatorics of the polytope created by taking the convex hull over all vector-parking functions. Last, we discuss planar tanglegrams, and provide an algorithm to generate them uniformly at random. An (m, n)-parking function can be characterized as . . . See PDF file for full abstract.CatalanCombinatoricsParking functionsTanglegramsGeneralized Catalan numberDissertation