  We study the growth order of the maximal displacement of branching symmetric
$\alpha$-stable processes. We assume the branching rate measure $\mu$ is in the
Kato class and $\mu$ has a compact support on ${\mathbb R}^d$. We show that the
maximal displacement exponentially grows and its order is determined by the
index $\alpha$ and the spectral bottom of the corresponding Schr\"odinger-type
operator.
