Put $K = |Types_{d,d}(Ball_r(v))|$. By the locality lemma, every node of $tree_{k,q}(v)$ is the tree of a tuple from the ball, and every color in a true atom of its root meets the ball. There are at most $K$ such colors and at most $K$ trees per rank and arity by part (i). So every root has $O_{k,q}(K)$ atoms, every node has at most $(q+1) K$ children, and the depth is $q$. This gives the bound $(3 + D) (K+1)^(q+1)$, where $D$ depends on $k$ and $q$ only.