Tight curves like that can be tough. First you will need to remove the old wood. Get some new lumber it looks like red oak from the picture. Get something with as tight and straight a grain as you can in my opinion it will work better. Next you need to cut Kerfs (Saw cuts) across the width of the wood where the curve will be.. The should be spaced about a 1/16" apart and be cut to a depth so you leave about a 1/16" material on the front of the wood. This takes a wile but can be accomplished using a table saw with miter gauge, a radial arm saw, or a circular saw. The kerfs will allow the wood to be flexible. I would also wet the face of the board with water when you bend it. Then using some glue fill in the kerfs and carefully bend it around the frame on the stair and carefully nail it into place preferably on any flat areas. This of course doesn't guarantee it won't break but it will help it. Hope this helps you out.