What size bucket? A five gallon bucket? A two gallon wash bucket? Ten buckets over a year doesn’t sound extreme, annoying but not extreme.
There is no excuse for anything to fall into his yard. 1. He owns the airspace over his yard all the way to the edge of space. It is a trespass. 2. It is very dangerous to have his property bombarded by falling objects. I would call OSHA and let them take on harassing the contractor.
If there is any damage to his tree where the beam hit it he has proof that they screwed up big time. Again, OSHA is his friend. They could have to write a new safety plan 5000 pages long for that one.