[eluser]WanWizard[/eluser]
What do you mean exactly by 'this library'? The only one I can find is the one in the first post, and that doesn't include a primary_key_column_name.
The parent_id is needed when you're using what is called the Modified Preorder Tree Traversal. So next to the left- and right pointers, you also have a direct link to the parent node in the tree. This can save some queries, depending on the implementation.