I found what I think is a persimmon tree in the park across the street. It's currently dropping small, orange-peach fruits all over the sidewalk - they look something like a plum, with six or seven flat, pumpkinish seeds inside each.
Having never learned the old adage, 'putting unidentified fruits in your mouth will get you an instant place in the Darwin Awards', I dissected and tasted a bit of one. It turned my mouth inside out like some sort of sensory Gestapo.
So yeah, I think they're persimmons. Said to be edible, but ... I think I'll wait for the local pecan crop instead.
Edit: At least the pecans are yielding well. I picked half a bag full of unripe ones off the ground, and there are still plenty more on the tree. The squirrels aren't going quite as crazy as the Leicester Chestnut Brigade, but it's early days yet.