a variant of the Kleene tree

Let $K_s$ be a computable monotone sequence of finite sets whose union is $K$, the halting set. Let $T$ be the tree of all $\{0,1\}$-sequences $\tau$ such that for some $s \geq |\tau|$, $\tau$ is the characteristic function of $K_s \cap \{0,\ldots,|\tau|-1\}$. The tree $T$ is of the type you want and its only infinite path is the characteristic function of $K$. $T$ cannot be a Kleene-type tree because of the Low Basis Theorem.