Tree Visualizations in Structured Data Recursively Defined by the Aleph Data Relation

In visualization, tree structures and parent/child data relations are basic. In an earlier article, the author demonstrated how a single tree structure emerged from relational data by recursion, and by the properties of the underlying strings of 0's and 1's of its data symbols. The paper takes a closer look at the components of this visualization to investigate this transformation in greater detail. The unifying factor in this transformation is Mark Burgin's mathematical theory of named sets. Applications of his modeling technique produce a system of models that scales from a basic data relation to a tree structure. The models themselves not only reveal a uniform pattern of data relations hidden in the RDBMS, they give shape and form to the recursive algorithm and to the data structure that organizes relational data into a monolithic tree. Therefore, this system of models represents one component of this novel visualization. The recursive algorithm that operates on these models represents a second. And the third component is the tree structure that this algorithm constructs that is called the open hierarchical data structure (OHDS). In visualization terms, all of three of these components are tightly integrated by the newly discovered data relation. Upon observing its formation, it becomes clear that this data relation represents a specific type of parent/child relation in structured data that is derived from a well-defined SQL SELECT statement. The paper identifies this data relation as the Aleph. In the database, its one-to-one and one-to-many branching patterns are self-similar to the tree structure. Until now, this uniform pattern was concealed within each database table, and in the table linkage of the database system. We conclude that the Aleph data relation is ubiquitous throughout the RDBMS. By the conclusion, the paper demonstrates that its tree visualization is mechanically constructed and recursively defined by the Aleph data relation. The discovery of the Aleph has both practical and theoretical implications that go well beyond visualization research.