Formulas and asymptotics of hypergraph Catalan numbers

Tree walks are a class of closed walks on a complete graph constrained to span trees. In this work, we focus on a special subclass called $k$-tours, which were recently introduced by Gunnells and are enumerated by the hypergraph Catalan numbers $c_n^{(k)}$. Gunnells conjectured an asymptotic formula for $c_n^{(k)}$ which we confirm through an alternative approach to their enumeration. As it turns out, the asymptotic growth is governed by the number of $k$-tours on star-like trees.