  For any compact oriented manifold $M$, we show that that the top degree
multi-vector fields transverse to the zero section of $\wedge^{\text{top}}TM$
are classified, up to orientation preserving diffeomorphism, in terms of the
topology of the arrangement of its zero locus and a finite number of numerical
invariants. The group governing the infinitesimal deformations of such
multi-vector fields is computed, and an explicit set of generators exhibited.
For the spheres $S^n$, a correspondence between certain isotopy classes of
multi-vector fields and classes of weighted signed trees is established.
