Trees that can be grown in "too many" ways: Bouch's construction and a bound for every size

We show that, for every positive integer $L$, there is a rooted tree on the square lattice that can be grown in at least $L!/C^L$ distinct ways, where $L$ denotes the number of bonds and $C>1$ is a constant independent of $L$. Bouch's hierarchical construction [Bouch2015] gives such trees along an u