Growing fruit trees from seed is a project that will take six or more years before any fruit would be produced. Fruit trees grown commercially are often the product of grafting seedlings onto an already fruit-bearing or dwarfing growth stock in order to produce fruit within the first one or two years. Rarely does a fruit tree grown from seeds of a particular variety of fruit bear fruit that exactly duplicates the fruit from which the seed was taken; any more than human children necessarily look exactly like their mother. Chances are that the fruit will have a mixture of qualities from both the fruit tree itself and the tree that pollinated the flower that produced the original fruit.
Cleaning and Drying the Seeds
Just for fun, seeds or pits taken from a piece of fruit from the supermarket can be cleaned to remove any bits of the original fruit and dried. To dry, lay them out on a flat surface and allow them to dry for a few weeks until the moisture is dried up, turning them a few times for complete all-around drying. Cracking the hard outer shell of a stone pit can be done to expose the seed inside in order to dry it, being careful not to injure the softer seed. After drying, the seeds can be stored in a ziplock bag or jar until ready for chilling.
"After Ripening" the Seeds
This is also called stratification and is done at the end of autumn or the beginning of the winter. Most fruit tree seeds need to be chilled in a packing material such as peat moss, shredded paper or sand, that can be kept moistened and occasionally vented throughout the chilling. The chilling induces a dormant state for three months which the seeds would need to germinate in the spring. This can either be done in a container filled with seeds and packing material in a ziplock bag in the refrigerator, or in a garden covered with sand and a wire mesh to protect from seed-eating wildlife and left in the cold winter temperatures outdoors. After the warm weather starts, the seeds can be taken out of cold storage and planted.
Planting the Seeds
The seeds, at the end of their stratification, are planted in small flowerpots or flats and kept moist (but not wet) under a tent of clear plastic. When the seeds begin to sprout, they are uncovered and placed in a sunlit area. They can be transplanted to a larger pot after the seedling grows to a height of 2 feet if desired. After they are 3 to 4 years old, the tree can be used as understock for grafting for earlier fruit production. Otherwise, they can be planted in the ground and allowed to grow into a tree that will bear fruit after 5 to 6 winters.

