Friezes, Triangulations, and Trees

Abstract: In this thesis, we focus on these three classes of objects: frieze patterns, polygon triangulations, and planar binary rooted trees. After proving that these objects are in pairwise bijective correspondence with each other, we introduce Catalan numbers through Dyck paths and prove that all these objects are Catalan objects. By studying the properties of triangulations and binary trees, we establish some properties of frieze patterns. In the final section, we provide a python code to generate frieze patterns together with their corresponding polygon triangulation. Additionally, we prove that the Pl¨ucker relations are satisfied in an arbitrary pattern.