If the cut off bits have a clean cut like a sharp scissors, that is rabbits. Ground squirrels or gophers are other possibilities. Gophers will come up out of their burrows at night and eat things.

You can put a cylinder of wire fencing around the tree for a while--that protects well. 2' tall is enough, unless your rabbits are jackrabbits then go taller. I pin down the cylinder with landscaping staples.

rats are a possibility too

Thank you both so much! All of these are possibilities. I will try the 2' guard. It is very hard to combat something that you can't catch "in the action".