I've looked in Ix and Alakol and some of Groddle Forest- but I haven't found a single one! Or are the trees supposed to be chopped down or something for them to become wood trees?
Wood trees can be planted (and sometimes found) in Ix, Alakol and Forest .. the "sometimes" is related to the fact that they are very short-lived - feel free to plant one if you come across a patch in any of these areas. You can also plant one in your own garden. Liberal doses of Fertilidust/Fertilidust Lite help them grow big big and thus acquire a bit more staying power in public areas.
Wood trees seldom last longer than it takes for one player to plant, raise and clear it. For some reason they weren't designed to grow very big so they can only be harvested a very few times. if they are harvested before they are fully grown then that is usually the end of the tree, I am not sure why there are Quests involving them. If you need one for a quest you will have to search for a patch and plant one. If you have a home in an area where you can grow a wood tree then that is the best way to go.
Some housing blocks have patches where anyone can plant or dig. I have seen them in Groddle Heights but there may be others as well.