Don't ignore the insurance company. You were driving his car with his permission. Your friend was a passenger. As an operator of your friend's car, you are likely covered under his insurance policy for the damaged tree. Your question raises the issue of your duty to cooperate with the insurance company as it conducts its investigation. You should return the adjuster's call and explain what happened. Another car was about to hit you, you swerved to avoid a collision and, unfortunately, hit the tree. If you and your friend fail to cooperate, the insurance may deny coverage for the claim.
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Your question needs some clarification. Is your insurance ("driver insurance") covering the loss or is the car insurance covering the loss?
In these types of situations, usually the insurance covering the car is "primary", meaning that the insurance from that covers the losses up to the policy limits, and the driver's insurance is "secondary", meaning that it only provides coverage if the primary policy is inadequate to compensate a covered loss.
My question is whether the car insurance is providing coverage and the $500 "surcharge" is actually your friend's deductible. If your friend's insurance is providing coverage for your liability, it may be a good idea to cooperate and provide a statement.
One twist may be if your friend's insurance is providing coverage but seeking subrogation or reimbursement from your insurance company. Have you reported the accident to your insurance company? If so, I would ask the adjuster in that case what position to take with regard to a statement.
This sounds like a single vehicle accident, but the question is, why did you swerve? If to avoid something there may not be any liability on your part. Since you are seeking coverage through your friend's automobile insurance policy, and you were the driver, you need to cooperate with the insurer. Whether or not cooperation means you have to provide a statement is a question, but the insurer could compel you to provide an examination under oath. At the very least, you need to file an operator's report with the local police agency, registry of motor vehicles, and the insurer of the vehicle if the damage exceeds $500.00.