I bet that if you use the outer ring as the back of the bow, you will get little to no heartwood in that size stave.
I love sapling bows. More than half of my bows are sapling bows. But when you're working with narrow diamter saplings, its best with woods whose sapwood alone is good for bow making (hickory, oak, plum and other friutwoods, HHB, buckthorn, cascara, etc).
Woods whose sapwood (alone) is poorer for bowmaking (e.g. Osage, Locust, Cedar, Juniper etc), are better from staves with a large enough dimater to support a good amount of sapwood.
I could be totally wrong, but if that mulberry tree is 4-5 inches in diameter, I'm betting you only get 2" of heartwood max. Without chasing a ring,(e.g. just using the outer ring as the back of the bow) you will only have heartwood in your handle, and little to none in your limbs.
Still may make a nice bow, but I think Blackhawks advice is sound.
Gabe