  In this paper we consider a large class of super-Brownian motions in
$\mathbb{R}$ with spatially dependent branching mechanisms. We establish the
almost sure growth rate of the mass located outside a time-dependent interval
$(-\delta t,\delta t)$ for $\delta>0$. The growth rate is given in terms of the
principal eigenvalue $\lambda_{1}$ of the Sch\"{o}dinger type operator
associated with the branching mechanism. From this result we see the existence
of phase transition for the growth order at $\delta=\sqrt{\lambda_{1}/2}$. We
further show that the super-Brownian motion shifted by
$\sqrt{\lambda_{1}/2}\,t$ converges in distribution to a random measure with
random density mixed by a martingale limit.
