A tree falling is normally considered an Act of God and no responsibility is attached to you. Your neighbor’s homeowner or auto insurance policy will cover the damage caused by your tree falling across property lines assuming your neighbor is properly insured.
An exception to this could occur if you know your tree is susceptible to falling because it is weak and shows signs of deterioration. If you know this or if your neighbor informs you that your tree poses a danger to his property, you could be determined to be negligent for not remedying the problem and thereby held liable. It is possible that your homeowner insurance policy could be called upon to provide coverage when you are negligent. This would result in a claim against you. If you are determined to have been negligent by not taking care of your tree, you could risk being non-renewed by your current insurance company. Finding a new homeowner policy could be difficult and certainly more expensive. So, if you know you have a problem tree, it would be best to take care of it before it falls and causes damage to your neighbor’s home or auto and be a good neighbor.