In most cases the homeowner’s insurance policy of the house that was damaged would pay, less your deductible. Falling objects are a covered peril.
But what if the tree is dead or diseased? Now the situation is different. You can ask your neighbor to remove the tree because you are concerned that it will fall and damage your house. If she refuses, you could send your neighbor a Certified letter and pictures documenting your concerns. Then if it does fall and damage your house, your insurance company has a much stronger case to “subrogate” back against your neighbor’s homeowner’s insurance and have them pay.
The best way to resolve this situation is to work with your neighbor. If you have a $500 deductible, that would be out of your pocket anyway, so you could offer to pay the amount of the deductible to share the cost of removal. That way neither of you have the hassle, potential bad feelings or the claim on your insurance.
If she still refuses to have the tree removed, in most states you have the right to remove any portion of the tree that is on your side of the property line. But before taking this action, check with your insurance agent and your attorney.