  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.
