What is the name of the database that satisfies this need?

Flat-file databases store data in a single table without any structural relationships between records. This lack of hierarchy means that each record is independent, lacking the parent-child relationships required by the question. Therefore, flat-file databases do not meet the need for organized subtrees.

Cloud databases refer to database services hosted in the cloud, offering scalability and accessibility but not inherently defining a specific data structure. The term "cloud" encompasses various database types, including relational and NoSQL, but does not specify the hierarchical relationship between records outlined in the question.

NoSQL databases are designed to handle large volumes of unstructured data and can support various data models, including key-value, document, and graph formats. While some NoSQL databases may implement hierarchical structures, they are not exclusively defined by the parent-child relationship required by the question, making them less suitable.