No you don't.

Lots of plants have leaves. A few don't, some primitive because they hadn't evolved them yet (e.g. algae) and a few because they lost them (broom, cacti). If there were no trees and nobody had ever seen a tree you could still explain leaves.

Lots of plants have wood. Things that aren't trees have wood. They're called bushes. Wood is a thing separate from trees. Not all trees have wood: bananas grow on really big herbs that people call trees because they are tree-sized, but they're herbs. Palm trees aren't really made of wood.