  In this paper we investigate the scaling limit of the range (the set of
visited vertices) for a class of critical lattice models, starting from a
single initial particle at the origin. We give conditions on the random sets
and an associated "ancestral relation" under which, conditional on longterm
survival, the rescaled ranges converge weakly to the range of super-Brownian
motion as random sets. These hypotheses also give precise asymptotics for the
limiting behaviour of exiting a large ball, that is for the extrinsic one-arm
probabililty. We show that these conditions are satisfied by the voter model in
dimensions $d \ge 2$ and critical sufficiently spread out lattice trees in
dimensions $d > 8$. The latter result also has important consequences for the
behaviour of random walks on lattice trees in high dimensions. We conjecture
that our conditions are also satisfied by other models (at criticality above
the critical dimension) such as sufficiently spread out oriented percolation
and contact processes in dimensions d > 4. This version of the paper contains
details not present in the submitted version.
