If you've grown a daylily chances are that you've noticed a new plant forming along the scape (flower stem) of one of your plants. The other day I came across one growing on a browning stem of one of my Daylilies. Hemerocallis growers call these plantlets that sprout from the stems "proliferations." Sometimes a proliferation can grow enough during a growing season to actually flower, most of the time they'll grow just enough for them to be harvested and planted. Proliferations are exact clones of the plant they are growing from and they're a great source of free plants.
Proliferations like these will continue to grow as long as the scape they are attached to remains green. As you can see the stem that my proliferation is growing on has already started to brown so chances are that I won't have enough time to harvest my plantlet. If you notice that your flower stem has produced a proliferation you can help it along by keeping the stem green as long as possible to help the new plant produce roots. You can do this by pollinating a flower (just like how you pollinate an Amaryllis flower) on the stem and creating a seed pod. The plant will keep the stem alive to feed the seed pod which gives your proliferation time to develop roots and survive on it's own.
If you're lucky and your proliferation has already produced roots before the flower scape has turned brown you can continue to let it grow. The browning of the stem will start at the top and work down towards the proliferation, right before the browning reaches the proliferation cut off the stem close to the proliferation and plant it in well draining soil. My flower scape is too far along the decaying process for this to work for me but I do have one more option. I can cut off the dead flower stem above the new plant and then cut the stem near the soil and place it in water and try to give it more time to develop roots.
Orchid growers have a different name for "proliferations" they call them "Keikis." Keiki supposedly is Hawaiian for baby or kid. The Keiki is left to grow on the flower scape of the orchid until it also produces a healthy set of roots. This naturally occurring method of asexual (no flower pollination involved) plant propagation can also be encountered by cacti and succulent growers. Recently I came across a blog entry by another garden blogger where she posted pictures of a new plant growing along the flower scape of her Haworthia fasciata. Whatever the name and whatever kind of plant you come across one on propagate it and get more plants. You can't ever have too many plants.