Putting rug on basement stairs


I've got an open, 3-stringer staircase leading to my basement. 12 stairs in total. 2 x 10 treads. There are 2x4 walls right next to the staircase. I used some ledger lock bolts to screw the outside stringers to the 2x4 walls. I am going to have them carpeted. I need to shore up the staircase by adding 3/4 plywood for the risers. My question is about the carpet and the tread material. If I carpet the stairs, do I need to change the treads to thinner material? My thought is the addition of carpet to a 2x10 tread will screw up the stair heights, especially on the first stair leading to the basement.
Thanks in advance for your advice.
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Millions or staircases have been carpeted with no change of anything. Not a big deal to just add the carpet.
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