  We study generalized branching random walks, which allow time dependence and
local dependence between siblings. Under appropriate tail assumptions, we prove
the tightness of $F_n(\cdot-Med(F_n))$, where $F_n(\cdot)$ is the maxima
distribution at time $n$ and $Med(F_n)$ is the median of $F_n(\cdot)$. The main
component in the argument is a proof of exponential decay of the right tail
$1-F_n(\cdot-Med(F_n))$.
