Just to clarify your problem:
- You have a Java application running on a server.
- The Java application is used to render web pages viewed in a web browser.
- You want to use jsTree, a Javascript library, to render pretty trees for your client to see.
- You want jsTree to show the structure of some folders on your server.
jsTree is a javascript library, thus, somewhere in your solution you have to execute Javascript. While it is certainly possible for Java to execute Javascript (Rhino looks promising), it would seem that the browser would be the most natural place.
How does jsTree take input? From the website, it indicates it supports Ajax and JSON. JSON can certainly be embedded in a page by a server app rendering the page, so that seems like a good fit.
So then a solution appears. Your Java server app renders the page. While it is rendering the page, it includes the jsTree library, and it embeds a JSON document containing the structure information of the server-side folders you want the client to see.
Javascript running in the client's browser feeds that embedded json data to jsTree, and you get a pretty diagram.