Well, OKC Master Gardeners tell me this tree is a Mulberry tree. " It is a Common Fig, Ficus carica, Mulberry Tree family " .

I've got no idea how this arrived in my neighbor's yard.

A bird planted it more than likely.

It's a Mulberry tree. You are wise to cut it down. You can use it to smoke with. It's sweet similar to apple wood.

Mulberries are one of these love/hate things. By your driveway, definitely no (because of the mess). Way at the back of a fairly large property, definitely yes (wildlife food). Looks like yours is interfering with the fence and causing a problem.