Some of the earliest database systems were based upon the Hierarchical Database model. This represented data in a tree like structure with parent and children, much like you are suggesting here. HDMS were largely superseded by databases built upon the relational model. The major reasons for this were that RBDMSRDBMS could model "many to many" relationships which were difficult for hierarchical databases and that RDBMS could easily perform queries that were not part of the original design whereas HDBMS constrained you to query through paths specified at design time.
There are still some examples of hierarchical database systems in the wild, particularly the windows registry and LDAP.
Extensive coverage of this subject is available in the following article