We are all aware that plants grow from seeds, but many varieties can be propagated. Yes, in the spring, gardeners split plants at the roots to thin out beds. But some plants can be propagated year round.
Propagation involves trimming a branch of a plant and putting it in water so that roots may form. Often, new roots form where leaves were formally. The Rex Begonia can be propagated in this manner.
Other plants, like the Spider Plant, grow pedicel leaders that produce both flowers (for seed reprodution) and plantlets. The plantlets look like mini plants with formed roots. One the plantlet grows large enough, it detatches from the plant to live as its own, self-sustaining plant.