Here are few to start with:

• Big-data: Social network graphs (such as facebook friends, linkedin connects): When one tries to work on friends of two different unrelated people, you are looking at two different rooted trees.
• Analytics: Decision-trees and random-forests are ways of finding optimal solutions to decision problems given few satisfying criteria. When one needs to decide if something is going to work or not (e.g. if a person is going to payback the loan given by a bank or not) - these random forests are one way of implementing that decision logic. In this case each branch becomes a deciding factor (such as is he employed, does he own car, is he married, is his bank balance >5000 USD etc...)
• Networks: Different domains in the internet. Each domain is a rooted tree with sub-domains as child nodes. In such cases, to build any network security protocols or to understand the basic network topology, one would be essentially dealing with the said forest data-structures.
• Big-data: Web-scraping and search-engines: When you are trying to index web-pages (for search, just like google), you would essentially end-up organizing all your crawler data in a forest type structure on your disks, with each website being the root of individual trees.

There are many other use-cases (such as database indexes, organization hierarchy charts so on.) - but the fundamental point is: a forest is just a tree without a common root.

Once you merge all the top-level roots of a forest and put a pseudo root, the whole forest becomes just one tree. This concept can be seen in our every day computers. 

Each drive or disk on our machines is a tree of folders/files. On Windows, all the C:\ and D:\ and E:\ etc. are trees, which essentially form a forest, but will have pseudo root called as "my computer" (so to speak). Similarly for unix.

Hope this helps.

Thank you,

GK (Gopalakrishna)

http://gk.palem.in