You can contact the property manager again, but you may need to bring a legal action, if the property manager and/or insurer refuses to pay. The issue will be whether  the property was  at fault in some way, such as by not trimming obviously dead limbs or cutting back sick trees, or because either employees or contractors under their control damage the tree or limbs in some way as to cause the limbs to fall. If the property was not at fault--say, for example, there was an unexpected strong windstorm that snapped off seemingly healthy limbs--then it wouldn't be responsible. You should consult with an attorney, who can evaluate whether you seem to have a case, what it might be worth, and also estimate what it might cost to pursue it, so you can make a determination of what to do. If you have collision insurance, you could also submit a claim to your insurer.