I posed a similar question to my agent when a neighbor's dead tree was in danger of falling on our property. When a tree or a branch falls and causes injury or damage, the homeowner's insurance normally covers it. If the tree or branch belongs to a neighbor and there was negligence (for example, they knew the tree was dead) then the neighbor's homeowners insurance covers it. If not, then the homeowners insurance of whose property it fell on covers it.
The best thing of course is prevention but many people - including landlord and neighbors - prefer to save money and let branches fall where they will so insurance will cover it. This is playing with fire though.
Likewise, attempting to prune back branches yourself from a tree that belongs to somebody else (neighbor, landlord) can make you personally liable as well if the tree or branches were to fall.
If your landlord is not responsive, you need to cover yourself just in case. Contact a local licensed tree company or arborist - they will do an estimate for free for pruning / cutting down the tree and tell you if the tree is dead or not (have them write this down). If it is rotted, send a copy of the quote along with a certified letter (w/ return receipt) explaining that a branch already fell on your car and that you want them to file a claim with their homeowner's insurance to reimburse you for damage.
This letter will protect you in the future in case something worse, God forbid, were to happen and you need to prove that the owner has been put on alert - in other words, he's been negligent.
Granted my agent also said to hire a lawyer but I found that just talking to my neighbor first and explaining the potential risk, and giving him a copy of the quote (keep original), solved the issue ASAP. I didn't even need to send the letter.