You can do better than that, cleaning a saw to the point where there is no debris on it from a recent cut is pretty difficult. I'm fairly sure my saw still has bits on it from trees in Canada and that's 15 years ago. This tree is pretty unique, you're going to have a very hard time proving that bits found that match the tree are from any other tree that you may have cut, especially if you can't show where you did that.
Sounds like the start of "The Wall", a series in which the remnants of a felled tree are found exactly over the border of England and Scotland, forcing an English and a Scottish detective to work together. The English one is a working mom with a deadbeat husband and a past of drug abuse, the Scottish one is an divorced, alcoholic middle-aged man with estranged children. Both have "come to grips with ..."