Tie the wire between the tree and the door in a way that the door opens away from the tree. The wire has to be as long as the distance between the tree trunk and the fully-open door. Tie the stone (as a weight) in the middle of the wire, so that it remains suspended above the ground even when the distance between the tree and the door is closest (when the door is closed). And, voila, your automatic door has been created.
