Measure the depth of the tread of the wooden stairs. Measure from the edge of the stair closest to you, or nosing, to the back edge of the tread, where the riser connects to the tread.
Cut a 1-by-1-inch pine board into sections using the measurement of the tread depth and a table saw. Cut two sections for each tread on your stairs.
Rub wood glue on two consecutive sides of one section of 1-by-1-inch pine.
Place this section of pine into the corner made where the bottom of the tread touches the stringer. Set the wood with the glued sides touching the tread and the stringer.
Drill two 1 1/2-inch wood screws into the 1-by-1-inch pine section -- the cleat -- through the top of the tread at 1 inch from the nosing and 3 inches from the nosing.
Drill two 1 1/2-inch wood screws into the cleat through the outside of the stringer at 2 inches from the tread nosing and 4 inches from the tread nosing.
Continue to add cleats to both ends of the treads until the entire stair has been secured with cleats.