Car damaged on landlord's property

[Pete Patrician]

I woke up this AM and heard some REALLY strong winds outside. I ran downstairs to move the car and found a large branch from a tall tree fell and took out my back window, and also dented the rear left quarter panel and left scratches and a chip in the paint. So far that's all the damage I can see.

I rent a room in a house. I was parked in the small lot behind the house. I let my auto insurance company know about this, but was wondering if this could/should be covered under the landlord's home owner's insurance. The tree is on his property and I've been told by some people that whether I was living there or just visiting, if someone's tree fell on my property their insurance would cover it. But I don't know much of this kind of thing to know if this will hold up in all cases. If it could be I'm sure it will be a hassle to make it happen, but I don't have much time as I have no rear window, need to drive the car, and it's snowing tomorrow.

If anyone has some knowledge to shed here please let me know.

PS. I don't have free glass coverage. If so I'd just get the window replaced and worry about the dent later. I'd just like to know if the landlord has ANY responsibility here at all.

Thanks!!