Are there trees for $(\Sigma^2_1)^{\text{uB}}$? If there is a proper class of Woodin cardinals, then  Woodin showed (using stationary towers) that $(\Sigma^2_1)^{\text{uB}}$ statements are generically absolute, where $\text{uB}$ denotes the pointclass of universally Baire sets of reals.
This generic absoluteness result has a more local version: if $\lambda$ is a limit of Woodin cardinals, then
$(\Sigma^2_1)^{\text{uB}_\lambda}$ statements are generically absolute for posets of size less than $\lambda$, where $\text{uB}_\lambda$ denotes the pointclass of $\lambda$-universally Baire sets of reals (or what some people would call $\mathord{<}\lambda$-universally Baire sets of reals.)
The "local" generic absoluteness for $(\Sigma^2_1)^{\text{uB}_\lambda}$ can be explained in terms of  trees (although the proof uses stationary towers instead.)
More precisely, let $\varphi(v)$ be a formula in the language of set theory  expanded by a unary predicate symbol. For every limit $\lambda$ of Woodin cardinals there is a tree $T_{\varphi,\lambda}$ such that in every generic extension $V[g]$ by a poset of size less than $\lambda$ we have
$$ V[g] \models p[T_{\varphi,\lambda}] = \{x \in \mathbb{R} : \exists A \in \text{uB}_\lambda\,(\text{HC}; \mathord{\in},A) \models \varphi[x]\}.$$
This tree is obtained from the scale property for the pointclass $\Sigma^2_1$ of the derived model of $V$ at $\lambda$.
Given these trees $T_{\varphi,\lambda}$, the "local" generic absoluteness follows by a standard argument using the absoluteness of well-foundedness.
My question is, can the "global" generic absoluteness for $(\Sigma^2_1)^{\text{uB}}$ also be explained in terms of trees, assuming that there is a proper class of Woodin cardinals?  More precisely, is there a single proper-class-sized tree $T_\varphi$ such that in every generic extension $V[g]$ we have 
$$ V[g] \models p[T_{\varphi}] = \{x \in \mathbb{R} : \exists A \in \text{uB}\,(\text{HC}; \mathord{\in},A) \models \varphi[x]\}?$$
I can think of two possible approaches, both with apparently serious problems.


*

*Consider the "derived model at $\text{Ord}$." Problem: this doesn't really exist.

*Define $T_\varphi$ as the amalgamation of the trees $T_{\varphi,\lambda}$ for various $\lambda$, e.g. all limits of Woodin cardinals, or all limit of Woodin cardinals above some point.  Problem: I don't see any way to show that the projection of such an amalgamated tree in some generic extension $V[g]$ is not too large.
 A: The answer is yes.  Hugh Woodin showed me the following argument, which I post here with his permission.
Let $\varphi(v)$ be a formula in the language of set theory expanded by a unary predicate symbol.  Given a pair of ordinals $(\alpha, \beta)$,
working in $V^{\text{Col}(\omega,\alpha)}$
we let $B$ be a universally Baire set of reals having Wadge rank $\beta$
in the model $L(B,\mathbb{R})$, which satisfies $\mathsf{AD}^+$.  Note that this model depends only on $\beta$ and not on $B$, and also that every set of reals in $L(B,\mathbb{R})$ is universally Baire because $B^\sharp$ exists and is universally Baire.  Let $T_{\alpha,\beta}$ be the tree of a $(\Sigma^2_1)^{L(B,\mathbb{R})}$-scale on the set 
$$\{x \in \mathbb{R} : \exists C \in L(B,\mathbb{R})\, (\text{HC}; \in, C) \models \varphi[x]\}.$$
By the homogeneity of $\text{Col}(\omega,\alpha)$ this tree is  is independent of the choice of generic filter and we have  $T_{\alpha,\beta} \in V$.
Let $T$ by the amalgamation of all the trees $T_{\alpha,\beta}$, so that $T$ is a tree on $\omega \times \text{Ord}$ and $p[T] = \bigcup_{\alpha,\beta \in \text{Ord}} p[T_{\alpha,\beta}]$ in every generic extension of $V$.
We claim that
$$ V^{\text{Col}(\omega,\alpha)} \models p[T]  = \{x \in \mathbb{R} : \exists C \in \text{uB}\, (\text{HC}; \in, C) \models \varphi[x]\},$$
for every ordinal $\alpha$. The right-to-left inclusion follows immediately from the definition of the trees $T_{\alpha,\beta}$,
so it remains to prove the left-to-right inclusion.
Let $G \subset \text{Col}(\omega,\alpha)$ be a $V$-generic filter and let $x \in p[T]^{V[G]}$, say $x \in p[T_{\alpha',\beta'}]$ for ordinals $\alpha'$ and $\beta'$.    We want to show
\begin{equation*}\tag{$*$}
\exists C \in \text{uB}^{V[G]}\, (\text{HC}^{V[G]}; \in, C) \models \varphi[x].
\end{equation*}
If $\alpha' = \alpha$, this is easy.
There are two remaining cases to consider:


*

*$\alpha' > \alpha$.

*$\alpha' < \alpha$.
In case (1), we have ($*$) by $(\Sigma^1_2)^{\text{uB}}$ generic absoluteness for $\text{Col}(\omega,\alpha')$.
In case (2), we use the fact that if 
$B \in V[G \restriction \alpha']$ is a universally Baire set as in the definition of the tree $T_{\alpha', \beta'}$, then $B^\sharp$ exists and is universally Baire, so there is an elementary embedding
$$ j : L(B, \mathbb{R}^{V[G \restriction \alpha]}) \to L(B^{V[G]}, \mathbb{R}^{V[G]}),$$
and we have $j(T_{\alpha', \beta'}) = T_{\alpha, \beta}$ where $\beta$ is the Wadge rank of $B^{V[G]}$. Considering the pointwise image of a branch witnessing $x \in p[T_{\alpha',\beta'}]$, we have $x \in p[T_{\alpha,\beta}]$ . Therefore ($*$) is witnessed by a set of reals $C \in L(B^{V[G]}, \mathbb{R}^{V[G]})$.
