This tree has green berries Sept 10th.
Do you know the name of this tree?
Nyssa sylvatica? If so, the fruit will turn almost black.
Here is a link that might be useful: black gum
Nyssa sylvatica has alternate leaves.
This is Chinese privet.
Forestiera ligustrina or similar sp.?
Here is a link that might be useful: Forestiera ligustrina
Well, I suppose that could be. A noticeable difference from privet is having the flowers/fruit in leaf axils along the branch whereas privet has them at the ends (terminal).
USDA does show it as being found in SC.