So I guess this is more like 'extending the olive branch' to keep things amicable.      

Every branch calls it something different.

The tenant had been complaining about a fear a branch from the tree might fall on her 5 pound dog. 

If the system is a central ducted one, then check to see if the branch lines have dampers.  

=)Sue K. hoa stories are legendary. i manage a property with a nazi hoa about the trees and landscaping. they literally waited around the parking lot and then chased me down when they saw me arrive to tell me to cut some branches....and proceeded to stalk me another time when i was selling a house in their community to get the owner to move a bag of mulch...but since i've managed the discussed property above, i haven't heard a peep from this one...., but i'll make calls tomorrow

In that case i tell them they immediately text a legible pic of the deposit slip as well as which branch the deposit was made in.  

You are from CT so read the Connecticut Judicial branch guide.

We have a Medical Teaching school here, University of Texas Medical Branch. 

Another non-real estate way to approach this might be to contact the administration department of that branch of the ministry service.

As long as there isn't any weeds or branches enroaching your neighbors property, they cannot cut down any trees or things of that nature.