Who is responsible for cleaning up a tree that has fallen?

Negligence

However, there is an "if" to that question as with many insurance issues. If the tree was normally healthy and the wind blew it down is the first area to consider.

1- Your insurance company would not pay for damages to the neighbor simply because you have no legal liability to the neighbor for the act of nature that caused the loss. Your first-party insurance is specific to you and your property, not his.

If the tree was obviously dead and you had prior notice from obvious observation or actual notice from the neighbor (usually by certified mail) of the dangers, you would have had a duty owed to the neighbor to abate that danger or hazard. Your failure to abate the danger would be the causal relationship to his damage.

2- In the above case, the insurance carrier would possibly pay for the damage to the neighbor out of the liability portion of your policy. The reason for this is the verbiage: "we will pay all sums you are legally liable to pay" in the liability section of the policy.

Disclaimer: This is general in nature and should not be used in any specific case or tort. Each carrier and each jurisdiction may have varying opinions in this area.

3. Furthermore, if your own insurer paid the claim resulting from the fall of a neighbor's tree, the payment would be subject to the policy deductible. The deductible is the amount set forth in the policy that you have agreed to pay toward the repair of a covered loss. In return for its payment, the insurer would succeed to whatever rights that you had to recover its payment, which is a process called subrogation. You are not a party to that action (unless you choose to be in order to recover your deductible), but you are likely to be a witness at any trial that is held.