A flatbow does not necessarily have a flat back. The name is derived from the flat(tish) cross section, as opposed to a round cross section (=longbow).
Any bow that is made from a (split) stave, will have a round ('crowned') back. The diameter of the tree determines the degree of curvature on the back. A very small diameter tree will have a high crown, and would not make a flatbow if the back was left round. However, it is possible to flatten the crowned back of a tree stave. That would be called a 'decrowned' stave. Decrowning is much like ring chasing, except you don't follow one ring across the width of the bow, but you follow a few, so the back will become flat. A board is always sawn flat. So a board bow will by default have a flat back (but it can still make a longbow).
Buy or borrow the book The Traditional Bowyer's Bilble Vol. 1. It will make everything crystal clear to you. Much more so than we can explain in a few replies here.