According to many sources (like the Indian Battlefield Tactics), Native Americans were fighting with tactics using (and adapted to) the grounds, and, whenever possible (e.g. in the woods) hiding behind trees and moving from one to another, using a horseshoe shape.
This has been used all over the world and throughout history, even if tactics evolved with time and weapons. They definitely were using trees, but I can't find any evidence that they were climbing them. Why wouldn't they, though, if needed?
And going unnoticed into an opponent camp at night is as old as warfare. It's been done by the حَشّاشِین -- order of assassins and by the ninja with their "tanuki-gakure" technique, intended to hide, spy or attack:
The practice of climbing a tree and camouflaging oneself within the foliage. Falls under "wood techniques" (mokuton-no-jutsu).
Any technique invented on a continent or by a group of warriors can also be invented somewhere else, like how hunting techniques are often similar and adapted to the country and grounds. So I guess that it's clearly possible that a Native American warrior would climb a tree for spying, hiding or trapping.