With the right wood you can achieve similar levels of strength/weight with wood as aluminum, but the volume is much bigger, so you often only see small aircraft made of wood. However, there are multiple issues of working with wood, the grain can significantly alter it's properties, only very specific species can be used, requires pieces to be glued together in a very specific manner and the process of validating it for aircraft use is very complicated as well.

Source: studied airspace engineering