  We prove that if G_P is a finitely constrained group of binary rooted tree
automorphisms (a group binary tree subshift of finite type) defined by an
essential pattern group P of pattern size d, d>1, and if G_P has maximal
Hausdorff dimension (equal to 1-1/2^{d-1}), then G_P is not topologically
finitely generated. We describe precisely all essential pattern groups P that
yield finitely constrained groups with maximal Haudorff dimension. For a given
size d, d>1, there are exactly 2^{d-1} such pattern groups and they are all
maximal in the group of automorphisms of the finite rooted regular tree of
depth d.
