  Elaborating on works by Abouzaid and Mescher, we prove that for a Morse
function on a smooth compact manifold, its Morse cochain complex can be endowed
with an $\Omega B As$-algebra structure by counting moduli spaces of perturbed
Morse gradient trees. This rich structure descends to its already known
$A_\infty$-algebra structure. We then introduce the notion of $\Omega B
As$-morphism between two $\Omega B As$-algebras and prove that given two Morse
functions, one can construct an $\Omega B As$-morphism between their associated
$\Omega B As$-algebras by counting moduli spaces of two-colored perturbed Morse
gradient trees. This morphism induces a standard $A_\infty$-morphism between
the induced $A_\infty$-algebras. We work with integer coefficients, and provide
to this extent a detailed account on the sign conventions for $A_\infty$ (resp.
$\Omega B As$)-algebras and $A_\infty$ (resp. $\Omega B As$)-morphisms, using
polytopes (resp. moduli spaces) which explicitly realize the dg-operadic
objects encoding them. Our proofs also involve building at the level of
polytopes an explicit functor from the category of $\Omega B As$-algebras to
the category of $A_\infty$-algebras, drawing from a result by Markl and
Shnider. This paper comes in particular with a detailed survey on operads,
$A_\infty$-algebras and $A_\infty$-morphisms, the associahedra and the
multiplihedra, as well as some details on the usual techniques used in
symplectic topology to define algebraic structures on geometrical (co)chain
complexes. It moreover lays the basis for a second article in which we solve
the problem of finding a satisfactory homotopic notion of higher morphisms
between $A_\infty$-algebras and between $\Omega B As$-algebras, and show how
this higher algebra of $A_\infty$ and $\Omega B As$-algebras naturally arises
in the context of Morse theory.
