  A spread-out lattice animal is a finite connected set of edges in $\{ \{x,y\}
\subset \mathbb{Z}^d:0<||x-y||\le L \}$. A lattice tree is a lattice animal
with no loops.The best estimate on the critical point $p_c$ so far was achieved
by Penrose(JSP,77(1994):3-15): $p_c=1/e+O(L^{-2d/7}\log L)$ for both models for
all $d\ge1$. In this paper, we show that $p_c=1/e+CL^{-d}+O(L^{-d-1})$ for all
$d>8$, where the model-dependent constant $C$ has the random-walk
representation $C_\mathrm{LT}=\sum_{n=2}^\infty\frac{n+1}{2e}U^{*n}(o)$ and
$C_\mathrm{LA}=C_\mathrm{LT}-\frac1{2e^2}\sum_{n=3}^\infty U^{*n}(o)$, where
$U^{*n}$ is the $n$-fold convolution of the uniform distribution on the
$d$-dimensional ball $\{x\in \mathbb{R}^d:\|x\|\le1\}$. The proof is based on a
novel use of the lace expansion for the two-point function and detailed
analysis of the 1-point function at a certain value of $p$ that is designed to
make the analysis extreamly simple.
