I posted a worried post on FB and a friend (thank you, Joe) suggested both that they might be soapberry trees and that I should take a piece of tree to a nearby nursery and they would tell me what it was. I did a little more research on the ever-so-useful interwebs and it looked very much to me like soapberries were a lot bigger than the berries on these trees, but I thought they might be hackberries. Probably the pecans were squirrel deposits.
Today I took a sprig of leaves and berries to the nursery and they said they are hackberries. Relief!