Wood with much grain run off is likely to fail .Some species are more tolerant than others.( Hickory, Hop hornbeam for eg.)
Staves are split from trees as you stated ,but not all woods require removal of the sapwood. Woods like Osage benefit from it but yew prefers to have a couple of rings of sapwood. Most "whitewood"staves (Ash ,Oak ,Hickory, Maple,Hop Hornbeam, etc.) are generally made with the layer under the bark as the back of the bow.
To confuse things even more , you could "decrown" a stave which kind of makes it a board . You could also use limbs and saplings. Some woods work better than others for this. For example , you would be hard pressed to find a Privet or Lilac big enough to call a log
You would likely get one bow from the trunk. Two if you were lucky. Both make excellent bows , however.
The grain on a bow ,you will find, is of the utmost importance. Why don't you try making a couple. No greater teacher than experience! You'll find lots of help on this site.