"He can probably cut the tree without asking pemission and dump the branches in your yard, but in doing so must not cause damage to other plants or buildings."

That doesn't follow. The neighbor, by US law, can legally trim/cut back any parts of a tree that overhang into his airspace- true. But he cannot pull more branches over to his side in order to cut them. He cannot go on another's property in order to cut more branches, including branches on the far side of the tree, which action is implied in the recounting.