Does anyone know if there is currently any research or any potential bounds on the number of trees on $n$ vertices with degree at most $3$? One can bound this above by $C_{n}$ the $n$th Catalan number, or $n^{n-2}$ from Cayley's formula, but I think I have found a way to bound it a little better. I just want to make sure I'm not overlooking something that has already been done.

Edit: Some motivation might be nice? I'm looking at these as dual graphs of triangulations of polygons. I'm trying to see if this could be used to establish a lower bound on the number of unlabeled triangulations of an $(n+2)$-gon.