Kotzer refers to cutting something off from its source of growth, and it includes harvesting grain, picking grapes, dates, olives, or figs, and pulling off any other fruit or branch from a tree. Cutting down trees to use for heating or building is included in this prohibition as well. It is also forbidden to pull grass out of a crevice in a wall or to remove fungi from bucket handles (Shabbat 73b and 107b). Taking an avocado pit or a branch out of water is forbidden as well, if it has started growing roots.
According to Torah law, there is no prohibition on picking fruits, branches, or leaves from a tree that has completely dried out. Since the tree is not absorbing nourishment from the soil, one who pulls off a part of it is not detaching it from the source of its growth. However, since this resembles harvesting, the Sages forbade it. (See SA 336:12.)
In contrast, if a branch was cut off a tree before Shabbat, since it is clear that the branch has already been detached from its source of nourishment, the prohibition of Kotzer no longer applies to it, and one may pull off its fruits on Shabbat. If it is a fragrant branch, one may pick twigs or leaves from it in order to smell them (Rema 336:8).