Consider Cantor space $2^\omega$ with the standard topology generated by open sets $[\sigma] = \{ \sigma^\frown x: x \in 2^\omega \}$. If $A \subseteq 2^{<\omega}$ and $x \in 2^\omega$, we say $A$ is *dense along $x$* if for every prefix $\sigma \prec x$, there is $\tau \succ \sigma$ such that all finite extensions of $\tau$ are in $A$.

An element $x \in 2^\omega$ is *1-generic* if, for every $\Sigma^0_1$ (computably enumerable) set $A \subseteq 2^{<\omega}$ which is dense along $x$, we have $x \in [A]$ ($x$ is a path through $A$). I think this is the standard definition (from here).

Now, suppose $T \subseteq 2^{<\omega}$ is a tree. What conditions can we impose on $T$ that guarantee $[T]$ contains a 1-generic member? Effectively, I'm looking for some type of "generic basis theorem". In particular, if $T$ is infinite and $\Sigma^0_2$, can we guarantee it contains a 1-generic path?