We did not get a survey. I regret it now, but it was not a requirement of our purchase (nor for title insurance purposes, etc) and it wasn't specifically recommended to us. That's why I'm not 100% certain if any part of the tree might already be growing on our property, and why I'm trying to plan out for both possibilities.

As far as cost honestly I would be willing to pay the neighbor extra up to double the cost of removal (as in he gets an extra amount equal to the cost of removal) or potentially even close to the replacement cost of the garage. I hope it wouldn't come to that, but it would certainly be less of a headache than getting the garage damaged, and if it makes him a happy neighbor.