What is the name of a dark purple fruit that grows on a tree?

I want to know the proper name of a purple fruit that grows on a tree. When I was a little kid all the other kids would pick them from the tree and eat them. They would also be a bunch of them on the ground as they had fallen. Every one called them 'Monkey *****' but I know that is not the real name. They look and shape exactly like purple egg plants but smaller I'd say about 2 inches long. They taste sweet but bitter at the same time if that make any sense. Do you know the name of this fruit? Thank you!