Question:

What is the best way to view Document Type, Subtype, and Classification hierarchy tree through the API?

Answer:

To obtain a combination of internal names of Type, Subtype, and Classification that exist in a Vault. All Subtypes of a Type and all the Classifications of a Subtype need to be used.

Example:

If Type A has Subtype B and C. 
and Subtype B has Classification D and E. 
Similarly, Subtype C has Classification F and G. 

Call all the combinations with the Retrieve Document Type/Subtype/Classification API Call:

A-B-D 
A-B-E 
A-C-F 
A-C-G 

Related Documentation:

Vault Retrieve Document Type API Documentation: Vault Help