I am reading "The Stationary Tower" trying to understand the proof of Theorem 1.5.12: "If $\delta$ is a Woodin cardinal, $Z$ is a set, and $T$ is a tree on $\omega\times Z$, then there is an $\alpha<\delta$ such that the forcing Coll($\omega,\alpha$) makes $T$ $<\delta$-weakly homogeneous."

It starts by showing that we can assume without loss of generality that $|T|\geq\delta$. Then it says "Fix a regular cardinal $\eta>\delta$ with $T\in V_{\eta}$ and let $T^{*}$ be the subtree of $T$ consisting of all nodes definable in $V_{\eta}$ from $T, \delta$ and parameters in $V_{\delta}$... $T$ and $T^{*}$ have the same projection in any forcing extension by a partial order in $V_{\delta}$."

I can see that this would be true if the set $Z$ had a definable well-ordering because then given any finite sequence of natural numbers $s$ there would be a definable node in $T_{s}$. But I am not sure why it would be true in general.