A tree in our backyard fell onto our neighbors fence...who is responsible for fixing the neighbors fence?

I am a lawyer in CT and practice in this area of the law.  A a practical matter, if the tree is on your property, you have a duty to keep it so that the branches do not get old and fall.  If you go to court and argue aver this, the court is likely going to award a judgment in the neighbor's favor since the tree in on your property.  I suggest telling the neighbor that you will split the cost to fix the fence.

If the tree fell because it was dead or dying, then you are liable for replacement of the fence and the removal of the tree.  See if your homeowner's policy covers this; it should (although you would have to pay the deductible).

If the tree fell due to an act of God (severe storm for instance), then you would bear no liability for any damages.  Your neighbor's policy would cover this.