From Barry Stone's "Ask the Inspector" column in the Real Estate section:
Question: The neighbors who live behind us have three large, dead trees that border our fence. A few years ago, our insurance company informed us that they would no longer insure our roof because it has wood shingles. If our neighbors' trees should fall on our roof, who would be responsible for the repair costs?
Answer: The exclusion of the roof by your insurance company may only involve fire damage or rain leakage. You should contact them to see if you would be covered for damages caused by a falling tree.
As for your neighbors, their dead trees constitute a significant public-safety hazard. Notify them immediately that you want these trees removed and will hold them legally and financially responsible for any personal injuries or property damage that may be caused by their failure to do so.
If they do not respond, the municipal authorities should be notified. The appropriate agency should then give notice to your neighbors to have the trees removed. If your neighbors still fail to act, the municipality can remove the trees and bill your neighbor for the work.
(Photo: Botanic Gardens Trust)