Different states have different rules regarding tenant and landlord responsibilities. That said, your landlord isn't automatically required to hand you money from their insurer after a tree falls and damages property unless they've agreed to pay for damage to your property. Even in that scenario, the payment method depends on the type of damage and how their insurer wishes to approve payment for the claim. In some cases, the insurer might only give you the money after you've purchased a replacement item or provide a damage estimate report from a recognized repair person or inspector. For vehicle damage, the insurer might pay your mechanic directly instead of you.
More importantly, insurance maintained by a landlord typically covers damage caused to property when other available insurance options don't apply to the situation. For example, your renter's or car insurance might come into play to help you recoup personal losses. Insurers look to the responsible party to determine who is at fault and who must pay. A number of scenarios impact liability: If your lease requires that you maintain the landscaping and a rotted tree fell, it's unlikely that a carrier would find your landlord at fault. If the landlord must maintain the landscape, then you could attempt to make a liability claim through the landlord's insurer. You must be able to prove though that the landlord acted negligently. Given the difficulties with proving such a claim, your best bet is to make a claim with your insurance company and then let them sort it all out.
As long as you personally aren't found to be at fault by having done something reckless, such as storing your motorcycle, boat, pool or other expensive item under a known rotted tree, your insurer will likely approve your claim and then chalk up the event to an act of nature or try to recover monies paid to you by addressing the matter with your landlord's insurance carrier. Keep in mind that if you believe your landlord is at fault and neither insurance company accepts your claim, you might have to take your landlord to court.