  We study the range $R_n$ of a random walk on the $d$-dimensional lattice
$\mathbb{Z}^d$ indexed by a random tree with $n$ vertices. Under the assumption
that the random walk is centered and has finite fourth moments, we prove in
dimension $d\leq3$ that $n^{-d/4}R_n$ converges in distribution to the Lebesgue
measure of the support of the integrated super-Brownian excursion (ISE). An
auxiliary result shows that the suitably rescaled local times of the
tree-indexed random walk converge in distribution to the density process of
ISE. We obtain similar results for the range of critical branching random walk
in $\mathbb{Z}^d$, $d\leq3$. As an intermediate estimate, we get exact
asymptotics for the probability that a critical branching random walk starting
with a single particle at the origin hits a distant point. The results of the
present article complement those derived in higher dimensions in our earlier
work.
