After planting, it will take 72 hours for the tree to reach maturity. Afterward, the tree will begin producing fruit, one fruit at a time, randomly. A pet can have up to three total apples on the tree at once, unless you have the Fruit Fairy, in which case you will produce up to four.
Birthday Cake fruits from this tree can be used to fish at the Pet Society Pond. So far, it looks like these fruits will only catch common fish and decorative / junk items.