re: Neighbor's tree fell on my fence today

In the state of Louisiana, you could file a claim against your insurance. In a windstorm event trees are not covered under a policy, if the tree falls on covered property (your fence) then the policy owes a reasonable amount to remove the tree from covered property (not the entire tree, just enough for the fence repair) and tree debris removal (most policies this limit is $500)

If the fence is a repair (less than a full run from end post to end post) then it is a repair and r/c is paid out. If the fence is a full run then the policy pays out depreciated amount (ACV) because the fence is not a building structure (roof and walls)

Your deductible then comes into play. You insurance company will subrogate against the neighbors insurance company. They will go against them for the tree removal only if the tree was healthy and go against them for the entire claim if the tree was dead.