Perfect subtree property for weakly compact cardinals

We investigate the consistency strength of the statement: kappa is weakly compact and there is no tree on kappa with exactly kappa(+) many branches. We show that this statement fails strongly (in the sense that there is a sealed tree with exactly kappa(+) many branches) if there is no inner model with a Woodin cardinal. Moreover, we show that for a weakly compact cardinal kappa the nonexistence of a tree on kappa with exactly kappa(+) many branches and, in particular, the Perfect Subtree Property for kappa, implies the consistency of AD(Double-struck capital R) + DC.