I would also suggest finding out if you are in a 'fence in' or 'fence out' county before dropping the money on a lawyer. 'Fence isn't he is required by law to keep his animals on his property, 'fence out' if you don't like them in your yard, it is your responsibility to figure out how to keep them out.
With 'fence in' you have all sorts of options, including legal action. With 'fence out' he may actually have the right to seek legal action against you for failing to keep his birds out of dangerous situations on your property.
Man am I glad I don't live in one of those backwards counties where I pay lots of money for my property, lots of money every year for taxes and insurance and then I'M the one responsible for keeping someone else's animals from damaging my property. Well chaulk one decent point up for New York State!