What would be the best way to implement a customizable (meaning, a tree structure with an unknown number of level) tree data structure in a database?
I've done this once before using a table with a foreign key to itself.
What other implementations could you see, and does this implementation make sense?
You mention the most commonly implemented, which is Adjacency List: https://blogs.msdn.microsoft.com/mvpawardprogram/2012/06/25/hierarchies-convert-adjacency-list-to-nested-sets
There are other models as well, including materialized path and nested sets: http://communities.bmc.com/communities/docs/DOC-9902
Joe Celko has written a book on this subject, which is a good reference from a general SQL perspective (it is mentioned in the nested set article link above).
Also, Itzik Ben-Gann has a good overview of the most common options in his book "Inside Microsoft SQL Server 2005: T-SQL Querying".
The main things to consider when choosing a model are:
1) Frequency of structure change - how frequently does the actual structure of the tree change. Some models provide better structure update characteristics. It is important to separate structure changes from other data changes however. For example, you may want to model a company's organizational chart. Some people will model this as an adjacency list, using the employee ID to link an employee to their supervisor. This is usually a sub-optimal approach. An approach that often works better is to model the org structure separate from employees themselves, and maintain the employee as an attribute of the structure. This way, when an employee leaves the company, the organizational structure itself does not need to be changes, just the association with the employee that left.
2) Is the tree write-heavy or read-heavy - some structures work very well when reading the structure, but incur additional overhead when writing to the structure.
3) What types of information do you need to obtain from the structure - some structures excel at providing certain kinds of information about the structure. Examples include finding a node and all its children, finding a node and all its parents, finding the count of child nodes meeting certain conditions, etc. You need to know what information will be needed from the structure to determine the structure that will best fit your needs.