Abstract: This paper is the 4th in a series of papers in which we propose a new finite element method for incompressible fluid dynamics-a dimensional reduction method. The divergence free space V^h is used as both the velocity solution space and the test function space in the momentum equation. Thus the pressure term disappear, the velocity vector can be solved(before the pressure).This paper presents a simple basis of V^h for a kind of first order finite element schemes solving three dimensional Navier-Stokes equations. It differs from the two dimensional problem on that the directly given "basis" B^h is linearly dependent. Therefore we must remove some functions from B^h so that it becomes linearly independent.All the removed functions form a "tree" in a sense.