|MadSci Network: Botany|
Your detailed description of the fruit makes me think that you must have a Red or White Mulberry. The fruit on a Red Mulberry are tastier to eat (Petrides et al, Eastern Trees)
As always, do not eat any fruit until you are sure of what you have. Get a book on trees out of the library and look up the Mulberries to check that you correctly identified this tree.
Hope this helps
Try the links in the MadSci Library for more information on Botany.