Inspect the carriage underneath the stairs, to ensure the treads and risers are flush with the carriage.
Drill a screw through the carriage and into each riser and tread at a 45-degree angle so they're flush with one another. Measure the wood and screws carefully, so the screws won’t go all the way through to the top side of the stairs.
Check the top of the stairs to see if the tread is still sagging. Add a wooden center support to the underside of the stairs to eliminate any sagging in the center of the treads. Cut the center support in a notched pattern to fit securely against the stairs, as the carriage does, from top to bottom. Drill one screw into each tread and riser at a 45-degree angle, the same as with the carriage.
Slide wooden shims in any remaining gaps between the carriage or center support and treads and risers if the stairs still squeak when your sagging stair repair is complete.