Of course, a simple small branch picked up from the ground (as long as it is not rotten) can because a wand simply by being in your possession.
If you know what kind of wood it is then even better, each wood has different proporties and are good for different kinds of spells and rituals.
You never need carving to make a wand, that is just an added extra that can personalise it and/or improve its proporties with sigils, runes, crystals etc.
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