We consider spread-out models of lattice trees and lattice animals on \({\mathbb {Z}}^d\) , for d above the upper critical dimension \(d_{\textrm{c}}=8\) . We define a correlation length and prove that it diverges as \((p_c-p)^{-1/4}\) at the critical point \(p_c\) . Using this, we prove that the near-critical two-point function is bounded above by \(C|x|^{-(d-2)}\exp [-c(p_c-p)^{1/4}|x|]\) . We apply the near-critical bound to study lattice trees and lattice animals on a discrete d-dimensional torus (with \(d > d_{\textrm{c}}\) ) of volume V. For \(p_c-p\) of order \(V^{-1/2}\) , we prove that the torus susceptibility is of order \(V^{1/4}\) , and that the torus two-point function behaves as \(|x|^{-(d-2)} + V^{-3/4}\) and thus has a plateau of size \(V^{-3/4}\) . The proofs require significant extensions of previous results obtained using the lace expansion.
