Short answer – not likely.  When your neighbors leaves or branches fall into your yard are they responsible to come rake the leaves in your yard?  No.  If your neighbor’s tree branches grow across your fence line are you permitted to trim them back so they don’t intrude into your yard?  Yes.
Most states, if not all, think of it this way.  When a leaf blows over into your yard – it’s your leaf.  When a tree branch or piece of fruit grows across your fence line – it’s your branch or piece of fruit.  And when or any part of a tree falls… the moment it crosses your fence line – it’s your tree.  And vice versa.
The question being asked is if the neighbor is liable for the damages to your home in a normal situation where their tree falls and damages your property.  If they had no knowledge that the tree was going to fall, and the tree fell through an act of God (wind, storm, lightening, etc.) then it is your insurance that will pay the loss – not theirs.
Why?

The usual way that their insurance would pay is if they were somehow liable.  This means that they likely knew the tree was going to fall or had fair warning the tree was going to fall and did nothing about it.  Let’s say that you wrote them several letters that the tree was in danger of falling and they did nothing about it.  Or maybe they were warned by the city and still did nothing about it.  Then possibly their liability insurance would kick in and compensate you for the damages or a portion of the damages.
However, even if you were able to prove liable, and ultimately their insurance company took responsibility to pay, you still may not have made the right choice by going through your neighbors insurance rather than your own.  Just like you, your neighbor has a contract with their insurance company.  The neighbor is obligated to pay their premiums and their insurance company is obligated to indemnify them for a covered loss under all of the terms and conditions of the policy.  However, their insurance company is not obligated to you – but rather to their insured.
In this situation it is often better to file a first party claim (you) with your insurance company.  Your insurance company is obligated and beholding to you to indemnify you for a covered loss with all the terms and conditions in force.  If you have one of the most popular homeowner’s policies, an HO3, one of those important conditions is that you are owed Replacement Cost.  Replacement Cost means the full value of the damages without deduction for depreciation.  Since their insurance company is not obligated to you – you may find that they only pay your Actual Cash Value which means full replacement cost less depreciation and in some states less taxes and contractor overhead and profit.
Therefore, the advantages of filing with your insurance company can be substantial.  Then let your insurance company battle your neighbor’s insurance company for any subrogation that may be available to them.  And some insurance companies have “good neighbor” procedures that allow for your neighbor’s insurance to pay your deductible or at least a portion of the deductible.
Contact one of our claim’s consultants if you have additional questions or need more information.