ls home/a/Downloads/t2/ from the directory you extracted the tar archive in.
Note how in the
tar t listing the paths don't have leading slashes, so they'll be taken relative to the current working directory when the archive is extracted, not relative to the filesystem root. Usually, that's what you'd want anyway, since extracting files from an archive to potentially anywhere could be rather dangerous.
Of course, when creating an archive, one often uses
tar cf foo.tar t2, and not
tar cf foo.tar /home/a/t2 to only store the relevant part of the paths.
But since you already have the longer paths in the archive, you could a) go to the root to extract it:
~$ cd /
/$ tar xf ~/Downloads/t1/test.t
or use an option to have
tar remove the unwanted parts, e.g. at least in GNU tar:
Strip NUMBER leading components from file names on extraction.
tar xf test.t --strip-components=3 should do.