Before doing anything, however, why not have a discussion with your neighbor?
If you have safety concerns, or if the branch is dead, there's a good chance you can come to an amicable agreement.
If communicating with your neighbor doesn't work and you want to remove the branch, your best course of action is to call a professional to do it.
Here is a list of people in Upstate NY charged with animal cruelty, according to the Mohawk Hudson Humane Society and the Albany County Sheriff's Department.