I am trying to construct an example of a infinite, finitely branching, recursive tree $T$ such that none of its paths has a graph which is $\Delta^0_2$. I denote the set of paths of $T$ by $[T]$. I have shown that if the tree $T$ is recursively bounded, then there is an $f\in [T]$ such that the graph of $f$ is $\Delta^0_2$. So my example needs to be finitely branching but not recursively bounded.
So far my attempt has been trying to mimic the the construction of a recursive binary tree such that no path is recursive. This construction proceeds by taking two disjoint recursively enumerable sets $A$ and $B$ so that there is no recursive set $C$ such that $A\subseteq C$ and $B\cap C=\emptyset$. Then one organizes the tree so that any path will be the characteristic for such a set $C$, whereby none of the paths is recursive.
Now we can certainly obtain two disjoint sets $A, B$ which are $\Sigma^0_2$ but cannot be separated by any $\Delta^0_2$ set by relativizing to $0'$. However, we cannot proceed to construct the tree in the exact same way as in the proceeding paragraph as then we would construct a binary tree, which is recursively bounded by the constantly $2$ function, and so would have a path whose graph is $\Delta^0_2$. So I need a more sophisticated construction, but I am not sure exactly how to go on. In fact, I am not even sure that proceeding in this direction is necessarily the best way to get my desired tree. Any indications on how to proceed with this problem would be greatly appreciated!