Hey Jimmy, Happy Birthday! Sorry to have to do that is THIS thread.
If it were me I would probably look up all your records for the work done on your clients tree, as well as the who,what when, where and why you worked on the complainants tree.
You never mentioned the species of tree that had the limb drop. By your explanation of the climb on the neighbours tree, I am assuming it was a codominat trunk tree.
What I meant by professional in my previous post was Assembling the records for the work done (including how it was paid for), species of tree, work performed (and why).... right down to the calls for the ticked off tree owner.
Getting all this done prior to some process server showing up, will get you ahead of the game, and the stresses involved in this type of thing.
It is probably wise to let the tree owner make the next move, as he has to prove YOU did something wrong! You don't have to prove your innocent. BUT I am NOT a lawyer, or even an American.
I do think a good defense is a good offense. There are thousands of reasons for trees and limbs to fail, it is up to the complainant to show why it failed.
Have a great B'Day, and don't sweat it. It will be there to deal with on Monday!