The young part of a plant that's almost ready to flower or unfurl new leaves is called the bud. As a verb bud also means to grow or develop.
You will see the word bud most often in reference to plants, but it is also used metaphorically to refer to something that is still maturing: a "budding career," for instance, or a "budding romance." Scientists use bud to indicate a new formation of growth or something that splits apart to form a duplicate of itself, like a cell.
