Falling trees limbs across property line
This is just a simple summary of what has been posted here.
Assuming the tree was not in an obvious bad state, then it's falling limbs resulted from an act of God. I am of the impression that in most states the removal of the falling limbs would not be the responsibility of the owner of the property where the tree trunk is rooted. Therefore, if the limbs fell on the neighbor's property, that neighbor could let them rot away or remove them at his/her expense.
In Tennessee, there were trees that were rooted in a neighbor's property, but in a storm fell onto my property causing considerable damage. Removal of the trees and the cost of damages was my and my insurance company's expense.
My insurance paid for fences and home damage, but not the removal or tree trunks lying on the earth. The insurance deductable was at my expense