You will need a court order to remove the trees. You will have to convince the judge that the trees are a nuisance preventing you from full use and enjoyment of your property, and that a trimming of the tree branches will not cure the problem. The court will balance the hardships - the detriment to you for allowing the nuisance to continue vs. the hardhip to the HOA if the trees were removed. You should start with having your attorney write a letter to the HOA where you offer to pay for the tree removal.
Richard A. Rodgers, Esq.
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It's not clear from your question whether you are a member of the HOA. If so, you need to take up this issue with your HOA board, not with the management company.
And even so, your HOA board may be limited in what it can do, depending upon whether or not your city or county has a tree preservation ordinance.