Need advice: Neighbors' dying trees

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My neighbors have two trees that sit very close to our property line, and they are not healthy. We have gotten a couple branches in our yard from them already, and on one it splits low at the trunk, and one half looks dangerously close to coming down. If it did it would land on our shed, and if, heaven forbid, the shed didn't stop it the next place of contact is my daughter's room. How do I go about getting them to take it down? They don't actually live there, as they are preparing for a sale. Should I wait until they sell it, and talk to the new owners? Is there any recourse I could take with the city to force them to take them down? Any advice/experiences would be appreciated!