Did your predator come back last night? Where is your pine tree in relation to your house? Do you have neighbors, dogs, activity that would deter a ground predator like a raccoon?
We had the same exact thing happen just today. It's extremely strange - I could have written your post myself - except I found the chicken this morning under our oak tree.
I'm guessing she'd been taken yesterday, but the description of the kill is almost the same. My hen still had her head, but her neck and crop had been removed. Only her windpipe and backbone remained. She was inside our fenced in yard - which means nothing really. Its not designed to keep predators out, only chickens in. But I'm positive she was killed yesterday because all the hens knew she was there. Once I let them out of the coop, they ran straight for her.
Our oak tree is right next to a chain link fence that separates our dogs (and our house) from our chickens. Our dogs bark pretty much all day (Beagles, you know) so I'm not inclined to think Raccoon. I was thinking Hawk or Owl.
What area are you in and what kind of predators do you have?