In the US, the neighbor in entitled to cut off any part of a tree that is over his yard. Imagine that the fence on the proprty line goes up to infinity. The airspace over the neighbor's property is his.

Call the police and have them take photos. Take you own photos of the tree and the debris he left.

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. If he did cause damage, sue him. If the tree dies he is in big trouble. Killing a tree on someone else's property can result in automatic "triple damages" in some areas. Our laws are based in English Common Law also.