Good morning..Let me see if I can give you some direction on this.
I will let you handle the niceties of opening and closing the letter, but here are the points you want to include:
1) Make note of the dead offending tree.
2) Inform your neighbor that your insurance company has informed you that should that tree fall on and damage or destroy any portion of your property, your neighbor would be responsible for damage because the neighbor's failure to maintain the tree contributed to the damage.
3) That you are hereby putting them on notice that the tree is dead and appears precarious, and that if not removed, in the event the tree causes any damage, you will hold them accountable.
I hope this has given you the guidance you were seeking. I wish you the best of luck!
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The information given here is not legal advice. As all states have different intricacies in their laws, the information given is general only. This communication does not establish an attorney-client relationship with you. I hope this answer has been helpful to you.