Tree Contractor Damaged Neighbors Home

[Mluna25]

Hi,
Happy Holidays to all.
I hired a tree contractor to cut down a beautiful, but sick (tree rot), Huge Red Oak Tree. I didn't feel it was safe to keep it up during the winter and was charged $7,500.00 for a terrible removal job.
The contractor not only left 4 feet of the stump, damaged my lighting system in my yard, but he also, in the process apparently damaged my neighbors home.
My neighbor did give the contractor permission to use her home to cut down the tree-as it oversees her property and I am in close quarters as we live in an attached row.
Her husband talked to the contractor at length over the phone and it seemed to be left that the contractor's insurance company would handle all the damages to the neighbors home, not to mine.
However, today, I received a letter from my neighbors insurance company expecting me to pay for ALL the damages-claiming that after receiving the facts they find me in fault?
I never spoke to the insurance company and from what I understand either did the contractor.
I am not sure how this works as I never had a problem such as this, but I was hoping to get a clear understanding of the liability in this situation.
Isn't the contractor responsible for paying for the damages since it was his workmanship that caused it?
The damage caused close to $20,000.00 or more-I am not sure...but I certainly do not have that kind of money. What are my legal rights and responsibilities in this case? Doesn't my neighbor hold some responsibility if she gave the contractor permission to work on her property? Or am I responsible for paying for this claim?
Any advise or suggestions would be greatly appreciated. I did contact my insurance agent and am awaiting to hear from them. But in the meantime, just want to know what the law states.
Thanks.
Troubled in NYC!

[Mr. Knowitall]

Put their insurance company in touch with yours, and both of them in touch with the tree removal company's insurance company, and let them battle it out amongst themselves.