What type of NoSQL database is best suited to store hierarchical data?
Say for example I want to store posts of a forum with a tree structure:
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MongoDB and CouchDB offer solutions, but not built in functionality. See this SO question on representing hierarchy in a relational database as most other NoSQL solutions I've seen are similar in this regard; where you have to write your own algorithms for recalculating that information as nodes are added, deleted and moved. Generally speaking you're making a decision between fast read times (e.g. nested set) or fast write times (adjacency list). See aforementioned SO question for more options along these lines - the flat table approach appears most aligned with your question.
One standard that does abstract away these considerations is the Java Content Repository (JCR), both Apache JackRabbit and JBoss eXo are implementations. Note, behind the scenes both are still doing some sort of algorithmic calculations to maintain hierarchy as described above. In addition, the JCR also handles permissions, file storage, and several other aspects - so it may be overkill for your project.
Exist-db implemented hierarchical data model for xml persistence
Here's a non-answer for you. SQLServer 2008!!!! It's great for recursive queries. Or you can go the old fashioned route and store hierarchy data in a separate table to avoid recursion.
I think relational databases lend themselves very well to tree data. Both in query performance and ease of use. With one caveat.... you will be inserting into an indexed table, and probably several other indexed tables every time someone makes a post. Insert performance could be an issue on a facebook caliber forum.
A NoSql storage service with native support for hierarchical data is Amazon Web Service's Simple Storage Service (AWS S3). The path based keys are hierarchical by nature, and the blob values may be typed using attributes (mime type, e.g. application/json, text/csv, etc.). Advantages of S3 include the ability to scale to both extremely large overall capacity, versioning, as well as nearly infinite concurrent writes. Disadvantages include no support for conditional writes (optimistic concurrency), or consistent reads (only for read-after write) and no support for references/relationships. It is also purely usage based so wide variations in demand do not require complex scaling infrastructure or over-provisioned capacity.
Check out MarkLogic. You can download a demo copy from the website. It is a database for unstructured data and falls under the NoSQL classification of databases. I know unstructured data is a pretty loaded term but just think of it as data that does not fit well in the rows and columns of a RDBMS (like hierarchical data).
Faced with the same issue, I decided to create my own (very simple) solution using Lua + Redis https://github.com/qbolec/Redis-Tree/