Using rational numbers to key nested sets

Abstract

This report details the generation and use of tree node ordering keys in a single relational database table. The keys for each node are calculated from the keys of its parent, in such a way that the sort order places every node in the tree before all of its descendants and after all siblings having a lower index. The calculation from parent keys to child keys is simple, and reversible in the sense that the keys of every ancestor of a node can be calculated from that node's keys without having to consult the database. Proofs of the above properties of the key encoding process and of its correspondence to a finite continued fraction form are provided.