  We study occupancy counts for the critical nearest-neighbor branching random
walk on the $d$-dimensional lattice, conditioned on non-extinction. For $d\geq
3$, Lalley and Zheng (2011) showed that the properly scaled joint distribution
of the number of sites occupied by $j$ generation-$n$ particles,
$j=1,2,\ldots$, converges in distribution as $n$ goes to infinity, to a
deterministic multiple of a single exponential random variable. The limiting
exponential variable can be understood as the classical Yaglom limit of the
total population size of generation $n$. Here we study the second order
fluctuations around this limit, first, by providing a rate of convergence in
the Wasserstein metric that holds for all $d\geq3$, and second, by showing that
for $d\geq 7$, the weak limit of the scaled joint differences between the
number of occupancy-$j$ sites and appropriate multiples of the total population
size converge in the Wasserstein metric to a multivariate symmetric Laplace
distribution. We also provide a rate of convergence for this latter result.
