Melanie, this is the paw paw that grows up here - native - IMPOSSIBLE to find for sale either @ native nurseries or online anywhere.
Slimleaf Pawpaw (Asimina longifolia)
It's hard to tell from your pics but I think this may be what you have down there. You would know better b/c of seeing yours in person. I think those leaves will elongate when they get more mature. If I recall correctly, that's what the ones I put in the db did - was started out more rounded & then elongated with maturity.
Charlotte, is it possible you could grow the Asimina triloba down on the riverbank?
