  This paper introduces an infinite linear hierarchy for the homogeneous,
incompressible three-dimensional Navier-Stokes equation. The Cauchy problem of
the hierarchy with a factorized divergence-free initial datum is shown to be
equivalent to that of the incompressible Navier-Stokes equation in
$\mathcal{H}^1.$ This allows us to present an explicit formula for solutions to
the incompressible Navier-Stokes equation under consideration. The obtained
formula is an expansion in terms of binary trees encoding the collision
histories of the "particles" in a concise form. Precisely, each term in the
summation of $n$ "particles" collision is expressed by a $n$-parameter singular
integral operator with an explicit kernel in Fourier space, describing a kind
of processes of two-body interaction of $n$ "particles". Therefore, this
formula is a physical expression for the solutions of the incompressible
Navier-Stokes equation.
