  The purpose of this paper is two-fold. First we explain the construction of
the canonical model of filtered $A_\infty$-algebras given in the authors' book
[FOOO]. The canonical model plays a crucial role in the study of Lagrangian
Floer theory on toric manifolds in our recent papers, arXiv:0802.1703 and
arXiv:0810.5654. Then using a variation of the arguments used in that
construction, we define a natural filtered $A_\infty$-structure on the Morse
complex of a Morse function and its $A_\infty$ homotopy to the
$A_\infty$-algebras on a Lagrangian submanifold constructed in [FOOO]. The
corresponding graphical moduli spaces `summing over trees' involve holomorphic
discs connected by the gradient flow lines.
