Keep in mind that there are generally three types of mulberry trees in the states. These are White, Red, and Paper Mulberry. Red is the only native mulberry, at least in the eastern half of the US. Unfortunately, one of these, Paper mulberry, is absolutely useless for bows, the wood being very weak and pulpy (I know - I cut a few down before I realized how weak the wood was). Honestly I would use pine before I would use this wood. There are some ways to seperate it from the other species, however. First, the leaves are fuzzy/pubescent (edible too), unlike the other two whose leaves tend to be smooth and glossy. Second, the buds are like little cones (think orange road cone shape) that tend to point away from the stem and are often opposite. The other two have buds that tend closer to the stem (specially in White, which clasps the stem), have the typical bud shape, and are never opposite. Third, Paper M bark tends to have more shallow furrows than the other 2 and can also have strange asymetrical brown patches here and there along it. It also tends to to grow in large patches of male trees which spread by the roots; If you see this you are very likely looking at Paper M.
Hope you've got one of the good ones!