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No you do not.
If the trees are on your land they are your responsibility and you have to ensure they are managed.
This involves trimming and root management where appropriate but if in a high wind bits fly off then this comes into act of god or S..t happens territory.
Firstly I always suspect anyone imposing a deadline, why tomorrow?, and secondly they have to show negligent management as opposed to just an act of nature.
If there are any further points please reply I will be happy to respond.
Kindly either accept my answer or rate it on the system so I get credit for my time.