The only scenario in which I could foresee an insurance company paying for the removal of trees (dead of otherwise) would be in the event that they were causing structural damage to the property (or other structure) through the spread of the roots. Even in this scenario, I imagine the any claim would be assessed on the scale of any damaged caused and how progressive the problem is.
As previously mentioned by others, an insurance company would not pay for maintenance or preventative measures (major incidents perhaps excluded). Furthermore, failure on the part of the homeowner to take reasonable steps to prevent a claim occurring may harm the final settlement.
I imagine that if you are enquiring as the whether you can make an insurance claim on such a thing that you are not in a favourable financial situation. Therefore, it is worth considering that, on the slim chance that it is covered, and you go-ahead and claim for this, how will the claim affect next year’s premiums or will you loose some kind of discount off you policy?
It is obviously advisable to consult any documentation relating to your policy as cover can vary from place to place and company to company.
In summery, I would say that it is best to only claim when you absolutely need to. Should you have to claim, consider the harm it may do to next year’s premium. If less people made frivolous claims the rest of us would not be burdened with high premiums.