So far I presented solutions where paths were computed when the code was executed. In the materialized path solution, the paths are stored so that they need not be computed repeatedly. You basically store an enumerated path and a level for each node of the tree in two additional columns. The solution works optimally with trees and forests.
This approach has two main advantages over the iterative/recursive approach. Queries are simpler and set based (without relying on recursive CTEs). Also, queries typically perform much faster because they can rely on indexing of the path.
However, now that you have two additional attributes in the table, you need to keep them in sync with the tree as it undergoes changes. The cost of modifications ...