I rent a house. Tree fell on my cars. Who is responsible?

My husband and I rent a house. For three months, we have had small limbs and branches fall from the trees lining the driveway. On multiple occasions, we have told the landlord that something needed to be done. He has always said he would look at it. Now, a huge portion of the tree fell out onto my car (05... show more My husband and I rent a house. For three months, we have had small limbs and branches fall from the trees lining the driveway. On multiple occasions, we have told the landlord that something needed to be done. He has always said he would look at it. Now, a huge portion of the tree fell out onto my car (05 Altima with insurance) and my husband's antique car (61 Chrysler no insurance). His homeowner's won't cover, our renter's won't cover. My auto policy will cover the Altima, but I will have to come up with the deductible and my rates will go up.

Is the landlord liable for any of the charges (including tree removal)? Please let me know any advice!

Thanks!
Update: The antique car was recently brought to us. It had been in another state. When we purchased our policies here, we did not have the car. And as you said, hindsight is 20/20. Tonight, I found out that antique insurance is roughly $50 a year.

Thanks for your input.
6 answers 6