Yes, it sure does. Giant Mushrooms lose their tops in the Winter but come back in the Spring. Don't chop it down, tap it instead! You'll get mushrooms every few days Spring thru Fall and of course a valuable Purple Mushroom once in a while. They will spread, with little mushrooms appearing nearby and growing into more giant ones. I love them!
You need untapped trees in fall, then there is a certain chance to get a mushroom tree. My farms are usually pretty wild with a lot of forests, so I always had at least one each fall.