The first method is "adjancency", a method that involves storing a parent ID in each node and recursing to find the related ones. He goes with a different solution, however - a "nested set". This method stores the data in terms of where it's located on the tree (layers from right, layers from left, etc).
Last night I was working on an application that display a full tree where the user must be able to to drag and drop nodes to reorder the tree. The problem is that it is much more difficult to insert or reorder the tree using the nested set model.