Koszul duality patterns in Floer theory

We study symplectic invariants of the open symplectic manifolds $X_\Gamma$ obtained by plumbing cotangent bundles of 2-spheres according to a plumbing tree $\Gamma$. For any tree $\Gamma$, we calculate (DG-)algebra models of the Fukaya category $\mathcal{F}(X_\Gamma)$ of closed exact Lagrangians in $X_\Gamma$ and the wrapped Fukaya category $\mathcal{W}(X_\Gamma)$. When $\Gamma$ is a Dynkin tree of type $A_n$ or $D_n$ (and conjecturally also for $E_6,E_7,E_8$), we prove that these models for the Fukaya category $\mathcal{F}(X_\Gamma)$ and $\mathcal{W}(X_\Gamma)$ are related by (derived) Koszul duality. As an application, we give explicit computations of symplectic cohomology of $X_\Gamma$ for $\Gamma=A_n,D_n$, based on the Legendrian surgery formula of Bourgeois, Ekholm and Eliashberg.