One-point recursions and topological recursion for enumerative problems

Many structures and shapes in nature exhibit recursion or self-similarity, in which large structures contain smaller copies of themselves at different scales. Topological recursion focuses this idea into a particular mathematical framework to analyse a large class of problems arising in enumeration, geometry and physics. This thesis explores a family of problems that fall under this umbrella. On the one hand, we enrich the general theory by adding new examples to the family; on the other hand, we develop a parallel theory of one-point recursions, which we conjecturally link back to topological recursion.