  We introduce a formula for translating any upper bound on the percolation
threshold of a lattice \g into a lower bound on the exponential growth rate of
lattice animals $a(G)$ and vice-versa. We exploit this to improve on the best
known asymptotic bounds on $a(\mathbb{Z}^d)$ as $d\to \infty$. Our formula
remains valid if instead of lattice animals we enumerate certain sub-species
called interfaces. Enumerating interfaces leads to functional duality formulas
that are tightly connected to percolation and are not valid for lattice
animals, as well as to strict inequalities for the percolation threshold.
  Incidentally, we prove that the rate of the exponential decay of the cluster
size distribution of Bernoulli percolation is a continuous function of $p\in
(0,1)$.
