Sometimes its the nails that are going up and down in the wood causing the squeak. I would consider screwing down the stairs since you are carpeting them anyway. Be sure to screw into a support board.
Look under each tread and see if you see any space. Then shim the area to stop the squeak. have an observer with a good flashlight looking at the squeaky tread real close as you walk on the tread and see if any movement is seen in each tread. If so, find out why. sometimes a good squirt of heavy oil on an area will stop it also.
This is really just an experimental task since you cannot look underneath.