There are a number of ways to propagate seedless fruits and vegetables, ranging from grafting to crossing specific seeded varieties to produce sterile offspring. Technically, the idea of a “seedless fruit” is a bit oxymoronic, as a fruit is, by definition, a fully matured ovary that contains seeds used for propagation. Given that such fruits and vegetables taste and feel like their seeded counterparts, however, use of the term is generally viewed as acceptable for the sake of convenience.
One method for propagating fruits and vegetables comes from the plants themselves. Many plants put out what are known as runners or offsets, which emerge from fully mature plants, and they are designed to spread the plant across a wider area. When a plant produces particularly tasty fruits and vegetables, these offsets can be encouraged to create a plantation, which is actually just a series of clones from a single plant.
Grafting can also be used to cultivate crops. This technique is most notably used with fruit trees, in which a branch from one tree is cut and attached to another fruit tree. If done at the right time of year when sap is running high, the graft will easily take, growing onto the parent tree and using that tree as a source of nutrition. With grafting, it is possible to grow a tree that produces a multitude of fruit varieties, including seedless fruits; grafting is also used to attach more fragile fruit trees to sturdy rootstock in cool climates.
It is also possible to grow seedless fruits through the use of cuttings, which produce clones of the parent tree. Cuttings are produced by snipping off sections of the plant and encouraging them to grow independently. Ultimately, the grafts will put out roots, allowing gardeners to plant them.
All of the above techniques for propagation have one serious flaw: they lead to a decline in biodiversity. Because they involve essentially making copies of one plant, if an agricultural disease that targets that plant evolves, it can spell big trouble. Many famous cultivars of seedless fruit, for example, are grown all over the world, and these stocks could be extremely vulnerable to disease or pests. The decrease in biodiversity is also bad for the species in general, as the more diverse a species is, the more likely it is to survive, as a general rule.
There is another method for breeding crops that involves crossing two varieties to produce a sterile hybrid. This technique is most famously used in watermelons. Essentially, two varieties with unequal numbers of chromosomes are crossed, creating an offspring that will produce fruit that never develop seeds. This technique tends to promote biodiversity, because each year, an entirely new crop is created, encouraging farmers to retain healthy stocks of seeded varieties to crossbreed.