Iirc, in fall, any untapped wild trees on your farm have a small chance each night to turn into one. (Obviously the fewer such trees you have, the smaller change you have to get one.) There's also a late game option to buy a mushroom tree seed.

Mushroom trees grow (if you planted a seed) and spread as usual for wild trees, and can be tapped for mushrooms. They go dormant in winter (look like a stump), but recover in spring if left alone. You can also chop them down for mushrooms (I've heard that chopping them down to a stump but leaving the stump will also recover in spring, but haven't tested it myself).