Carbon monoxide (CO) forms when there isn't enough oxygen to form carbon dioxide (CO2). If you are in an open environment, CO poisoning isn't a concern. If you are in a closed environment (i.e. - running a car in the garage w/ the door closed) CO poisoning is a definate problem because the hemoglobin in the blood prefers to pick up CO over oxygen (which is missing from the area anyway) - before long, you nod off and never wake up because your blood has no oxygen in it.

But I digress - how would you put a wood burning stove in a tree? they are HEAVY!!! plus you would have to haul wood up! (unless you burn the tree you are in one branch at a time!)