It's good that the neighbors' cats help with the rodent population, but I know that it's a concern with a dog that uses the same yard.
Our chicken yard tree is a big maple.
That's true about coyotes. We don't see many of them anymore. There were over a dozen when we first moved here.
We have to worry about foxes (I was shocked to learn that our foxes climb, and nest in, trees!), raccoons, etc. Before I changed my chicken coop design, and changed the wire, we lost chickens to them, and once even to a bear. :-( Even with the change, we had that bobcat here. A lot of predators.
Hopefully. She isn't too thrilled when Dh plucks her off of the fence after dark. Sheesh.
Love All Life, Thank You For Posting! :)