What are parts of organisms growing into whole ones?

Answer 1

FragmentationMain article: Fragmentation (Biology)

Fragmentation is a form of asexual reproduction where a new organism grows from a fragment of the parent. Each fragment develops into a mature, fully grown individual. Fragmentation is seen in many organisms such as animals (some annelid worms and sea stars), fungi, and plants. Some plants have specialized structures for reproduction via fragmentation, such as gemmae in liverworts. Most lichens, which are a symbiotic union of a fungus and photosynthetic algae or bacteria, reproduce through fragmentation to ensure that new individuals contain both symbionts. These fragments can take the form of soredia, dust-like particles consisting of fungal hyphae wrapped around photobiont cells

Source: Wikipedia

http://en.wikipedia.org/wiki/Asexual_reproduction#Types_of_asexual_reproduction