"AWT" stands for "Ancestry World Tree", a collection of user-submitted files now displayed in a very similar manner to those in WorldConnect since the latter became part of the same conglomerate. It claimed (in January 2013) to contain "nearly 400 million names in family trees submitted by our users. The Ancestry World Tree is the largest collection of its kind on the Internet."
AWT can often fill in a gap in a genealogy, but is generally no more reliable than any other user-submitted site.
The same phrase "Ancestry World Tree" is apparently not "trademarked" and may be seen in advertisements for OneGreatFamily.com, which in January 2013 claimed to have "over 236,779,785 submitted pedigree-linked names".