When a tree falls into a house it is covered by insurance 99% of the time. It does not matter if it was your tree, the neighbors tree, or a city tree, it is your insurance that pays the claim. I know this sounds absurd, but it's been the law of the land for ages and there's no point in trying to push a rock uphill. It might have been your neighbors tree, but you and your insurance company are stuck with it.
The only exception is if you have given the property owner from where the tree stood, written notice, via certified mail, and included a certified arborist report specifying the tree hazard. Then the property owner or their insurance must pay the claim.
Many insurance companies strive to pay out as little as possible on claims. Remember the adjuster might act like your friend but works for and is paid by your insurance company. Seldom are they structural engineers knowing what is necessary for repairs to be performed up to code. Their assessment of your damage is not final, it is the very first assessment.
Many adjusters' initial figure is often less than half of what is necessary and actually paid in the end. It is your job to present documentation as to exactly what is required to make your house whole again. Do not trust what the adjuster sees shining his flashlight in the attic checking for damage. Hire a reputable construction firm to give you a written estimate for repairs. Insurance companies cannot ignore detailed bids from professional contractors.
Yes, there are many good insurance companies and adjusters but there are also those who act as though they have final say on your claim. This is absolutely NOT TRUE.