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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 
$d$, 
$d\geq 2$, 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 Hausdorff dimension. For a given size 
$d$, 
$d\geq 2$, 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 
$d$.
