I've just finished my 10th tree, so I can safely say nothing happens to your streak once you finish a tree. The only time something happens to your streak is if you miss a day, you will lose it.
People with multiple hundred day streaks (like you, awesome!) should by the Streak Freeze from the Lingot Store so that they don't get frustrated if they miss a day?
Thanks! It's always a good idea to have a Streak Freeze anyway, regardless of how long your streak is. You could be away on vacation, a power outage could occur, etc. and you don't want to lose your streak, as it can be a great motivator!
I know.......i lost my 80 day streak and for about a week i just didn't feel like doing duolingo...EDIT-And i forgot to buy a streak freeze ;(