I've been doing some testing on the Tree of Prosperity. Once you buy it, you can imbue it to make it harvestable straight away. If you do that, you get to pick which fruit you want - if it's the first time, you can pick any of the four (Harvest, Mind, Soul or Body). If it has already been harvested before, you can't choose the last one picked. You don't have to imbue the tree, it will come to life on it's own after about 4 hours of being picked. When it comes into bloom on it's own, the fruit is random and changes every four hours.
When you pick and eat the fruit you gain skill. The amount you gain varies, I gained anywhere from less than a full skill point to 2 whole skill points at times.
I tested for about 4 Sim days with 6 trees and these are the skills you can gain. There may be more skills, but I didn't see any others.
Fruit of Harvest (green) - cooking, fishing and gardening
Fruit of the Mind (blue) - charisma, inventing, writing, handiness and logic
Fruit of the Soul (purple) - writing, painting, sculpting, photography and guitar
fruit of the Body (pink) - athletic, handiness, martial arts and riding