Trees that can be grown in "too many" ways: A review of Bouch's construction

Abstract

We carefully review the hierarchical construction by Bouch [Bouch2015] of trees on the square lattice that can be grown from its root in $L!/C^L$ distinct ways, where $L$ denotes the number of bonds constituting the tree, and $C>1$ is a constant. (As discussed in Section IV.A of [ParkerCaoAvdoshkinScaffidiAltman2019] and Appendix A.3 of [ShiraishiTasaki2024], this result has an implication on the operator growth in quantum spin systems in two or higher dimensions.)