Whether your own tree is threatening to fall on the neighbor's house, or he is making grumbling noises that he is afraid it may fall on his house, or whether the leaf is on the other branch so to speak, get an expert, preferably a consulting arborist to evaluate the situation. Is the tree diseased, dead? Over-topped and dangerously leaning? Are the roots exposed, undermining the sidewalk? Get a report in writing to show the neighbor.
2. Talk to the neighbor
Report in hand, open the discussion in a friendly manner, offer a glass of wine or tea and cookies. Show him what the problems are and ask if he will be willing to help deal with it. If he's the one making the grumbling noises, assure him you will take the necessary steps to alleviate his concerns.
3. If he won't listen, stay calm
Remain calm, explain your concerns again. If he insists your tree is a pain in the tuchas to him, no matter your reassurances, you don't want him taking self-help action. Many a favorite old oak or maple has gone under the midnight axe, or succumbed to deadly toxins, which, surprise, your neighbor has no idea where they came from.
4. Offer to mediate
Will he mediate? Many towns have tree committees or informal boards to hear such disputes. If not, offer to pay for a professional mediator. He can get his own expert and you can try to reach a compromise. Remember you have to live here for the foreseeable future.
Offer to share the cost of remediation if you reach a compromise, even if he is to blame. If not, take responsibility graciously.
5. If mediation fails
Now is the time for lawyers. First a polite letter, outlining the expert's findings and the applicable law. Is there a tree ordinance in your town? Find out what it is, what and who it protects. Is your tree a heritage tree, or an undesirable species. These can vary widely. Is there a view ordinance protecting the neighbor's right to restore a pre-existing view. Again, steps 1 - 4 can go a long way to avoiding costly litigation, which, if all else fails, you may find yourself mired in.
Additional resources provided by the author
Check my blog: Land Use News for articles and tales from the front.
Check the Facebook Pages: Green Legal Solutions , Tree Mediators and Urban Forestry Associates for the expert advice or a consulting arborist. Again, all in California. I cannot make any representations of the law or give advice for those outside the State.