It all depends on why the tree fell in the first place.
If it fell on your fence because your neighbor was negligent, then it would be the neighbor's responsibility. However, you may have to PROVE your neighbor was negligent in some way and that can be difficult to do.
It's never as simple as it seems.
And regardless of whose insurance pays, you should probably be aware that the details will vary by state. For instance, even the broadest forms of homeowners policies in Virginia typically depreciate for fences. (Even if you have full replacement cost on the structure itself and/or replacement cost on contents.) So, you may want to check on that before you decide to file a claim on your own policy -- it may not be worthwhile, depending on the amount of damage (because you'll be charged claims points if you use your insurance to pay for the damages and that means you'll pay higher insurance premiums for the next three years.) Definitely worth it for big claims, but rarely for small ones.
Talk to your agent. Ask for the specific clauses of your policy that apply and talk to them about whether or not the value of your fence would be depreciated. At least you'll know what you're dealing with going in.
licensed agent: life & health, property & casualty