For those who do not know, this poster is referring to a staircase where the treads are slipped in from behind into what is called a housed stringer. The entire width of the tread is supported on both sides and in the middle by the riser, which is also housed into the stringers.
Here is a good description. Read Chapter 5, page 60.
http://books.google.com/books?id=ycAD5m ... Y#PPA60,M1
But to answer your question, you will need to trim flush to the stringers, the front part of the tread that overhangs, called the nosing. It needs to be trimmed flush to the sides and the riser. At that point, you may need to tighten things up with some screws. Also, the remaining section of nosing that is INSIDE the housed stringer will show somewhat AFTER you install your new nosing or tread. I recommend before installing new material, that you sand smooth this area and use a wood patch/filler to match this spot as smooth as the rest of the stringer. Hopefully, your's is paint grade. If not to be painted, You may want to consider veneering it with 1/4" hardwood plywood BEFORE you install the new material.