You are concerned, but no damage has occurred
A useful first message identifies the observation and the location: a newly visible split above a shared path, for example, rather than “your tree will fall.” Offer existing photographs and ask whether an arborist can assess it. A professional report can clarify the tree decision, but it is not a legal finding that the neighbor is negligent or must pay for your preferred work.
Notice next: Keep a dated copy of your communication and any response. Ask a local adviser how to proceed if a serious concern remains unresolved.