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$\boldsymbol{d}\boldsymbol\gt \textbf{8}$
A spread-out lattice animal is a finite connected set of edges in 
$\{\{x,y\}\subset \mathbb{Z}^d\;:\;0\lt \|x-y\|\le L\}$. A lattice tree is a lattice animal with no loops. The best estimate on the critical point 
$p_{\textrm{c}}$ so far was achieved by Penrose (J. Stat. Phys. 77, 3–15, 1994) : 
$p_{\textrm{c}}=1/e+O(L^{-2d/7}\log L)$ for both models for all 
$d\ge 1$. In this paper, we show that 
$p_{\textrm{c}}=1/e+CL^{-d}+O(L^{-d-1})$ for all 
$d\gt 8$, where the model-dependent constant 
$C$ has the random-walk representationwhere 
\begin{align*} C_{\textrm{LT}}=\sum _{n=2}^\infty \frac{n+1}{2e}U^{*n}(o),&& C_{\textrm{LA}}=C_{\textrm{LT}}-\frac 1{2e^2}\sum _{n=3}^\infty U^{*n}(o), \end{align*}
$U^{*n}$ is the 
$n$-fold convolution of the uniform distribution on the 
$d$-dimensional ball 
$\{x\in{\mathbb R}^d\;: \|x\|\le 1\}$. The proof is based on a novel use of the lace expansion for the 2-point function and detailed analysis of the 1-point function at a certain value of 
$p$ that is designed to make the analysis extremely simple.
