  We study lattice trees (LTs) and animals (LAs) on the nearest-neighbor
lattice $\mathbb{Z}^d$ in high dimensions. We prove that LTs and LAs display
mean-field behavior above dimension $16$ and $17$, respectively. Such results
have previously been obtained by Hara and Slade in sufficiently high
dimensions. The dimension above which their results apply was not yet
specified. We rely on the non-backtracking lace expansion (NoBLE) method that
we have recently developed. The NoBLE makes use of an alternative lace
expansion for LAs and LTs that perturbs around non-backtracking random walk
rather than simple random walk, leading to smaller corrections. The NoBLE
method then provides a careful computational analysis that improves the
dimension above which the result applies. Universality arguments predict that
the upper critical dimension, above which our results apply, is equal to
$d_c=8$ for both models, as is known for sufficiently spread-out models by the
results of Hara and Slade mentioned earlier.
  The main ingredients in this paper are (a) to derive a non-backtracking lace
expansion for the LT and LA two-point functions; (b) to bound the
non-backtracking lace-expansion coefficients, thus showing that our general
NoBLE methodology can be applied; and (c) to obtain sharp numerical bounds on
the coefficients. Our proof is complemented by a computer-assisted numerical
analysis that verifies that the necessary bounds used in the NoBLE are
satisfied.
