  Geometric structures on $\mathbb N Q$-manifolds, i.e.~non-negatively graded
manifolds with an homological vector field, encode non-graded geometric data on
Lie algebroids and their higher analogues. A particularly relevant class of
structures consists of vector bundle valued differential forms. Symplectic
forms, contact structures and, more generally, distributions are in this class.
We describe vector bundle valued differential forms on non-negatively graded
manifolds in terms of non-graded geometric data. Moreover, we use this
description to present, in a unified way, novel proofs of known results, and
new results about degree one $\mathbb N Q$-manifolds equipped with certain
geometric structures, namely symplectic structures, contact structures,
involutive distributions (already present in literature) and locally conformal
symplectic structures, and generic vector bundle valued higher order forms, in
particular presymplectic and multisymplectic structures (not yet present in
literature).
