  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 $AD_{\mathbb{R}} + DC$.
