  We provide asymptotics for the range R(n) of a random walk on the
d-dimensional lattice indexed by a random tree with n vertices. Using Kingman's
subadditive ergodic theorem, we prove under general assumptions that R(n)/n
converges to a constant, and we give conditions ensuring that the limiting
constant is strictly positive. On the other hand, in dimension 4 and in the
case of a symmetric random walk with exponential moments, we prove that R(n)
grows like n/(log n). We apply our results to asymptotics for the range of
branching random walk when the initial size of the population tends to
infinity.
