  By introducing a new measure for the infinite Galton-Watson process and
providing estimates for (discrete) Green's functions on trees, we establish the
asymptotic behavior of the capacity of critical branching random walks: in high
dimensions $d\ge 7$, the capacity grows linearly; and in the critical dimension
$d=6$, it grows asymptotically proportional to $\frac{n}{\log n}$.
