Stair tread damage - repairable/replaceable? (photo)
If it's easy to remove the tread (and it may be) it's likely easiest to just replace it.

If it's not so easy to replace it, what you could do is put in a couple of pocket screws using a pocket screw kit. Kreg is a popular brand that makes pocket screw kits that you can find at most any big-box store. 

You'd set up the jig, drill your holes, squirt a bit of wood glue into the crack, then screw it together. 

You then fill the screw holes by gluing the included wood plugs in, give it a sand, and ready to paint.