Select a natural wood color acrylic caulk that matches the wood of the stairs and is designed to bond with wood.
Apply a thin bead of caulk along the length of the gap.
Wipe up any excess caulk with a scrap of rigid cardboard. Clean up residue by wiping gently with a damp cloth or paper towel.
Allow the caulk to dry according to the manufacturer's directions.
Brush out as much dirt as possible from gaps that are more than 1/4 inch wide with a clean, dry paintbrush.
Cut a piece of rope that is approximately as wide as the gap, to the same length as the gap. Press it into the gap using the handle of a small paintbrush or other tool.
Apply matching wood stain to the rope, working slowly so the stain has time to soak in.
Measure the thickness and length of the large gap in the stairs.
Remove the tread by inserting a pry bar below the nosing extension -- the overhang on the tread -- and pulling gently upward.
Measure the depth and width of the tread, and add the thickness of the gap, to determine the dimensions of the replacement tread.
Set the table saw's gauge to the size of the new tread. Adjust the blade's tilt to round the edge of the nosing to match the other treads.
Sand the nosing of the cut tread with 100-grit sandpaper, matching the profile of the other treads.
Stain or paint the new tread and allow it to dry completely.
Apply construction glue to the bottom of the new tread and to the tread's location on the portion of the staircase frame that will support the tread.
Press the tread into place so that it butts against the riser, or vertical panel, behind it.
Hammer several small finishing nails into the tread several inches back from the nosing and fill the nail holes with a matching filler stick.