For your protection please Never just assume anything as far as what can be done, check all ordinances, and if you are not sure of their interpretation have a professional interpret them for you. The cost to replace a tree that has died as a result of an improperly amputated limb/ radical pruning will blow your mind and I am not exaggerating.
Most likely you have the right the cut the branches that overhang the plane, if removing those branches will not kill the tree and a professional decides that. Do Not Cut/ Prune before you have all of your bases covered, as you may not be able to recover financially from taking the bull by the horns/ tree by the limb if the tree dies.
You need to check ordinances and I would imagine the HOA also has specific verbiage in regards to procedures and there are probably two sets of rules for trees planted and wild trees (trees that were there before the people were).
This is what I know as far as tree stuff blowing, be it leaves, debris, small little branches ~ nuisance tree debris that comes into your yard and clogs gutters, walkways, or swimming pool in this instance are considered just that, a nuisance, more specifically your nuisance. Even if the tree in which they fall from is on an adjoining lot it's an act of nature and you are responsible for cleaning up after mother nature. Mother Nature does not observe property lines as it relates to nuisance debris. Damage is another story but when there is damage it will have to be proven that there was neglect on the tree owner and their neglect caused your damage. Now your ordinances could be different as well as HOA. So that's something to look into as well. HOA may specifically address nuisance debris and be more strict, and that may fall in your favor.
It's worth a check for sure.