A comparison of definitions of equivariant trees

We show that various categories of trees can be modeled by Grothendieck constructions on categories of trees with a fixed set of leaves. We prove this result for the dendroidal category $\mathrm{\Omega }$, the category ${\mathrm{\Omega }}^G$ of trees with a $G$-action for a finite group $G$, and finally for the category of genuine equivariant trees ${\mathrm{\Omega }}_G$ that has played an important role in recent work on genuine equivariant operads.