Aurum trees only generate in the Flower Forest biome. Their leaves sometimes contain gold nuggets which drop when harvested.
Naturally generated Aurum Trees in a Flower Forest biome. A patch of Yellow Orchids can be seen in front of them.
Edelwood Trees only naturally generate in Dark Forest biomes. Their unique appearance consists of a hollow log that is up to four blocks high. The top part always has a face carved in it. On the side up to two Edelwood Branches might generate. When destroyed these Branches will drop an Edelwood Stick.
Smelting an Edelwood Log will result in the creation of Dark Matter.
A three-block high Edelwood Tree with two Edelwood Branches on the sides.
From time to time Edelwood Logs might get oily. You can collect this oil using an empty Glass Bottle and craft it into Black Dye.
Two normal Edelwood Logs on the left and two oily Edelwood Logs on the right.
The small leaves that the Carved Edelwood Log has can be removed by using a shear on the block. If you want them back you might want to use Bone Meal on the log without leaves.
Left: Carved Edelwood Log with leaves. Right: Carved Edelwood Log with leaves removed using a shear.
Alternatively to finding Edelwood Trees you can also create your own. Everything you need is a Dark Soul and an Oak Sapling or a Dead Bush. Simply right-click the Sapling or Dead Bush with the Dark Soul and it will transform into a Growing Edelwood. After some time this special kind of sapling will grow into a full Edelwood Tree.
It's a decoration for the moment that you can place by right-clicking (it may have a utility in a future update).