Are you solely responsible to pay for damages if a tree in your yard falls into your neighbor's yard and damages his fence?

No. Mother Nature would be the "responsible" party. The homeowner is not "liable" for acts of nature.

Your homeowners insurance is not responsible for a natural act that causes damage to property of another. Your neighbors fence is not listed as covered property on your Homeowners policy.

If a tree falls on your neighbors property the neighbors insurance coverage would invoke, it does not matter where the tree came from. Likewise if a tree falls on your property due to a natural occurrence your own homeowners insurance policy would cover you.

Answer

If the tree was healthy and you had no complaints from your neighbor and a wind storm came up and blew it over you may have a nice enough neighbor that will share the cost of a new fence.

AnswerAccording to my insurance agent, so long as there is no negligence on the part of the tree owner it is an "Act of Nature". Thus the fence owner would need to pay to make repairs if they want it fixed. My insurance covers me if my neighbor's tree falls on my house. I can't imagine the company paying for anything that they could make someone else pay for.