I have two Paw Paws in my yard, both are from the same “grove” of Paw Paw suckers in northern Illinois. Both pretty much died back to the ground this past winter. Now they are growing well from shoots that came out near ground level. I would like them to eventually become trees rather than bushes. Am I wise to let them grow as much as possible this year, them cut back the shoots so only the largest, most upright of the two shoots remains? Or should I cut back the lesser shoot now?

And, can anyone identify what kind of Paw Paw they are from the picture? Thanks!