Neighbors liability of tree branches over my house Sacramento CA

[Derek Daun]

I just got a call from my tenant informing me that very large tree branch from the neighbors tree fell on the top of the house.  I haven't gotten chance to inspect it yet, but there is likely to be some shingle and drywall damage. 

This leads me to a couple questions:

• Is the neighbor's insurance liable even if the tree branch was hanging over my property before it fell? I know in California I have the right to trim the branches (with some limitations) that come over my property, but do I have the obligation?
• This tree is directly next to the property line, such that trimming all the branches over my house would take off half the branches. Could this limit my obligation/right to trim it? The case law regarding limitations on trimming your neighbor's tree seems ambiguous, for example you can't tree the trim if doing so could harm the tree. 
• Should I preemptively contact my insurance company? They did a thorough inspection before insuring the property, but I'm concerned they might second guess that decision if I'm liable of the branches over my house.

Some more background information: This is a hundred+ year old neighborhood, that back in the day, decided planting giant cedar tress in the front of the houses was a good idea. These trees are massive: 60-80 ft tall, trunks 5 feet in diameter, branches the size of trees. One such tree is in my neighbors front yard, about 10 feet from either of our houses. When I moved into the house as an owner occupier 5 years ago, the tree was extremely neglected, with many dead branches. We had some work done on the tree on our side of the house, basically removing all the dead branches. The owner of the house next door was not in any condition to cooperate with us at the time. (Original owner passed away, daughter moved in and promptly picked up a meth habit). Since then the house was flipped. We've had multiple conversations with the current owner regarding the state of the tree, so she is aware it is an issue, but never anything in writing.