You may be able to have your neighbor pay to fix your house. However, and as a default rule, you should immediately put your homeowner's insurance carrier on notice of the damage caused to your house because you may have insurance coverage for such a peril, which you will need if it turns out that your neighbor is not legally responsible for your damage. Remember, you must document the damage with pictures and fill-out any required insurance claim forms prior to even thinking about repairing the damage to your house and garden.
Failing to provide timely and appropriate notice is a common reason that insurance coverage is denied. In fact, your insurance policy may require you to provide for an inspection of your property damage and also for you to answer questions, under oath, prior to commencing any repairs. Alternatively, if continuing damage is resulting from the opening caused by the tree (i.e., you are now experiencing water damage), you likely have an inconsistent duty to immediately fix the damage. Read your insurance policy carefully.
Now, turning to your potential claim against your neighbor, in states like New York, the determination as to whether you have a claim depends on the condition of the tree prior to its fall. If the tree was left in a state of decay and that condition caused its fall, then your neighbor is liable to you for the cost of repairing the damage to your house and for the removal of the tree from your property coupled with the damage to your garden, grass, etc. that the tree's removal caused.
However, if the tree was in a sound condition prior to its fall, you have no claim against your neighbor available. In fact, in determining whether the tree fell from decay or natural causes, there was no prior duty on your neighbor to consistently and constantly check all of her trees for nonvisible decay before the tree fell. Only where the decay was readily observable prior to the tree falling will your neighbor be liable to you.
As a result, when trees fall from natural cause, such as a hurricane or tornado, your neighbor is not responsible for the damages caused by the tree or even the removal of the tree itself. Instead, your neighbor, in such natural circumstances, only has an obligation to remove the tree to the extent that it exists within her boundary line after the fall.
If you believe that you have a claim, immediately call an arborist or a horticulturist to document and evidence the condition of the tree prior to its fall. Next, get in touch with an attorney to ensure that you have all of your ducks in a row prior to asserting your claim.
Adapted from this Dan's Papers post.