I spent several years in the Corps before coming over to the Army. I still do pullups every other day to maintain and to help me in my MOS (Infantry). I would recommend doing the following which has never failed me. Do a set of pullups (doesnt really matter the reps) as long as they are strict form. Do a set of pullups immediately by a set a pushups. Do this for 3 or 4 sets every couple of days. That way you keep improving and working both, this will help your back for ropes climbs and a variety of other exercises.
Personally, I do to tri-set (pullups followed by pushups and dips. For BCT the main three exercises mentioned, but you will need a strong back and core for roadmarches, rope climbs, etc. Push yourself!! PM if you want other exercise ideas.