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I'd go for a spyderco knife, they are small, have high performance stainless steels that you can leave in a bucket of water, and not cumbersome due to size allowing for finer work (still won't beat a wood carving knife).
If you want to cut down trees, I would suggest just buying a tungsten carbide, or a diamond impregnated wire.
Primitive people fair well with just knapping obsidian, or flint. If you want to split logs, you can just use a rock and a wedge like a cave man.