1) for example try visiting : http://en.wikipedia.org/wiki/Architecture_(disambiguation)
-->Is there any wikipedia api via which I can access this?
2) for example try visiting : http://en.wikipedia.org/wiki/Architecture When you do right at the bottom of the page, there is a Categories: Architecture Architectural design Greek loanwords
-->Is there a way to get this data? Preferably in form of a tree?
You could use the MediaWiki Query API; although it doesnt return trees or something similiar complex. You'd have to structure it yourself.
For example, you could get the list of all pages in the Category:Disambiguation pages and get all the links on each page in that list.