I think it is most likely Chinese privet - Ligustrum sinense, yes.
Alabama and other states are also being invaded by
Japanese - L. japonicum and Glossy privet - L. lucidum.
There are eradication programs; folks can report the location of the plants. You can rip them out, or burn them, or spray them with nasty chemicals.
http://www.issg.org/database/s...
