A tuple $v̄ ∈ X^ℓ$ is padded to a $d$-tuple by repeating its last entry. The type of quantifier rank $d$ of the padded tuple determines the type of $v̄$ of quantifier rank $d$, which determines $tree_{k+i,q−i}(v̄)$ by $tree_eq_of_tp_eq$, since $ρ⁺(k+i, q−i) + q − i ≤ d$. So the type trees are the image of a function on $Types_{d,d}(X)$.