You are essentially dealing with two separate but somewhat linked issues: a) would anyone be happy with having to clean their car everyday because of bird poo? b) what is the legal situation and your rights?
The answers are a) not in the slightest, it does seem excessive but there are possibly ways round this (car cover, parking in a different space etc) and then b) you've got no real legal rights in the situation that I am aware of, the landlord is due you a parking space that you've paid for - this has been provided. If you can prove that your car will be permanantly damaged by the bird poo and that the landlord or owner of the tree owes you a duty of care then you might just have a case but frankly I'd expect it to be laughed out of court when there are other solutions without resorting to the legal system.
//edit as for damaging your car, that is different but you would need some way to prove how it happened and who caused it. It doesn't necessarily have to be an admission of guilt from them but you would need some other form of evidence in the absence of an admission.
