Next to plants wood if one of the most important aspects of creating a naturalistic vivarium. It is great for the animals to climb on and hide under. Moss and epiphytic plants will grow on it, and vines will use it for support. It is important to choose the wood carefully, as the wrong wood my be harmful or not hold up well in the tank. Thankfully today there are many options choose from.
Cork Bark: A common accent wood, cork is often used in the vivarium hobby. flat sheets of it are great to use as a background for the vivarium. Usually, though, it comes in curled/curved pieces with are great as hides. It is also very light weight.
Ghost Wood: This is a beautiful light colored, smooth wood. Moss loves to grow on it, and it holds up very well in humid environments.
Grape Wood/Vine: It will not hold up as long as many of the woods on this list but it can work in a frog tank as long as it is not kept very wet. This means don't use it for tanks with water features and that get misted more than once a day. It has a great look in a natural vivarium.
Malaysian Drift Wood: This is a heavy, dark wood. It is one of the most popular woods in the hobby for it look and durability. It can even be used submerged under water.
Manzanita: This is a strong wood but it looks delicate. It is light in color and often comes with many long branches that are great for climbers. When used upside down in a tank it gives a branching root system look.
Mopani Wood (Also called Swahala): This wood comes treated (sealed) and untreated. Use the treated for dry tanks. For wet tanks only use the untreated (Some suppliers will call it "Safe for aquarium use").
If you are adventuresome you can use wild wood, though I don't recommend it since you might not know where it has been. But if you get it from deep enough in the wood then it should be fine. Some wild (I know all wood comes from the wild but you get what I'm saying) woods that hold up in wet environments are: elm, beech, and oak. Wild wood should be soaked in a dilute bleach water solution overnight, then soaked in plain water overnight, and then dried at 200 degrees in an oven for a few hours.
If you will be using your wood with a water feature you might want to leech out excess tannins. Tannins are found in the wood and can make your water dark, as well as lowering the pH. The animals usually don't mind but clearer water looks better. To leech out the tannins soak the wood in clean water (Changing the water everyday) until the water is almost clear. I say "almost" clear because you will not need the water to be 100% clear since the filter can handle the rest.
Most wood will mold when put in a tank. don't worry about it. You can scrape it off and eventually it will stop coming.
You can buy Mopani wood using the link below.