OK, I see the motivation.
*You could occasionally have a branch (or wood) on the ground near trees. I guess branch would need to be the same as wood in that case.
* If daggers and swords could be cutting instruments, they could be used to make wood. But that doesn't make sense for chopping down trees. And then if you added a crafting tier for cutting - that doesn't allow chopping trees down - nah, i think that would be just confusing.
* maybe any weapon usage on a tree has a chance to spawn a branch/wood. That would alleviate the need to have to fuss about with cutting property.