I actually did make a sword model, I just didn't make any animations with it haha
Everything that gets affected by the wind subscribes to a global wind system that feeds them a wind direction and strength. They handle them in different ways, the trees specifically use them in a shader that does some vertex movement with it. It's just a slightly modified version of this GPU Gems chapter about vegetation in Crysis. Instead of using the Y value of each vertex to determine movement amount though, I paint all of the vertices based on how much I want them to move with the wind (black being none, white being fully affected). The base, for example, doesn't move at all, so it's painted black. This technique is really helpful for low poly art since I can manually prevent the bundles of leaves from moving too far off the branch.
I'll clean up the wind system and put it on GitHub for reference at some point, but hopefully, that article helps in the meantime!