This is a tree that my father grew from a cutting 4-5 years ago. It has various shaped leaves as seen in pics. The fig is a breba that isn't fully ripe.

Eric, an image of the outside of the fruit would be helpful.  It appears to be green, correct?  There's not too many figs that have fruit that is green on the outside and yellowish on the inside so that would help narrow it down.  However, you say it isn't fully ripe.  Do you know if the pulp stays that color when it is fully ripe?  You might compare the fruit and leaves to images of fig varieties shown here:

http://figs4fun.com/Varieties.html