  Given a commutative algebra $\mathcal O$, a proper ideal $\mathcal I$, and a
resolution of $\mathcal O/ \mathcal I$ by projective $\mathcal O $-modules, we
construct an explicit Koszul-Tate resolution. We call it the arborescent
Koszul-Tate resolution since it is indexed by decorated trees. When the $
\mathcal O$-module resolution has finite length, only finitely many operations
are needed in our constructions -- this is to be compared with the classical
Tate algorithm, which requires infinitely many such computations if $ \mathcal
I$ is not a complete intersection. As a by-product of our construction, the
initial projective $\mathcal O $-module resolution becomes equipped with an
explicit $A_\infty$-algebra.
