Sweep the staircase with a broom to remove as much dust and debris as possible. If the existing stair treads extend past the riser--the vertical board in front of each step--measure the riser's length and transfer this to the riser.
Aim a laser level across the stair tread, lining it up with the mark you made in the previous step. Cut the stair tread's overhang off, following the laser line, with a reciprocating saw. Repeat with each stair tread.
Remove any uneven parts from the stair tread's front edge with a 1/2-inch chisel and a hammer. When you are finished, the stair tread should be flush with the riser. Repeat with the remaining stair treads. Sweep up the debris with a broom.
Measure the existing stair tread's depth from front to back,on both the right and left side. Lay a grain tread face down on a flat surface and transfer the measurement you just took to the tread. Measure back from the lipped edge, which will be the tread's front edge when installed. Mark the tread's left and right sides.
Lay a level on the grain tread and line up its edge with the two marks you just made. Use the level as a straight edge to draw a line along the tread. Cut the grain tread along the line with a table saw.
Set a stair measuring tool on the first step's existing tread. Open the measuring tool until both ends rest on either skirt board. Tighten the tool's wing nuts to anchor the tool at the appropriate length.
Center the stair measuring tool on the grain tread. Trace along the tool's ends with a pencil to mark where to cut the tread. Cut the tread along the two lines with a sliding miter saw.
Test fit the grain tread over the existing stair tread. If it fits correctly, apply a bead of construction adhesive along the existing tread's surface. Cover as much of the tread's surface area with the adhesive as possible.
Set the grain tread on the existing tread. Push down on it so it makes firm contact with the existing tread. Secure the tread in place with four finishing nails, one in each corner, from a pneumatic nail gun.
Repeat the process of adding grain tread to each stair tread until you have completed then grain treading process.