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I was a liability insurance claims adjuster for 11 years before I changed careers. You are late in the game to be looking for money now. If the tree was healthy and a storm blew it down it is an act of God and no liability involved.
If the tree was dead and you own the fence. You could make a case for the damage to your fence because the tree owner was negligent in not taking the dead tree down knowing it was a hazard. You would need some type of proof ( photo's ) that the tree was dead but still upright prior to you asking your neighbor to take it down before something happened.
As it stands, you appear to have no proof if the tree was alive or dead when it fell on your fence.
Unfortunately, in Georgia - it's your problem. I've been down this road twice. A lawyer may have a different view since the fence was damaged (and he wants to get paid). Google "Georgias fallen tree law"