Our five story ash tree has two main trunks; one with a cavity at the bottom. Over the last few years we have taken out several major branches which encroached our neighbors or overhung our house and also to lighten the crown and allow more light penetration. Some of this work was done last summer. Suddenly this spring two major branches turned up completely dead, one in the center of the crown and one to one side. Our usual arborist gave us quotes on entrenchment pruning and removal. Having done some research on entrenchment, we're not sure this is the way to go and we don't want to lose the tree. It provides great shade to the house and abundant habitat for bird and other life. Seeking your advise.