Concentrate mainly on the three you mentioned but pull-ups will not hurt. So long as training at pull-ups does not detract from pushups, situps, or running, I say have at it. All it will do is make you more fit.
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.
When I started training for pullups, I went from barely getting one rep out, to being able to consistently knock out sets of ten. When that happened, I found my upper body strength was worlds away from what it was, despite previously being able to max my pushups. It definitely helped out in all other exercises requiring upper-body strength. It's definitely an important exercise, that although it's not given that much importance in the APFT, should be used frequently in every soldier's fitness training plan.
with the new PRT you will do pull ups and variations thereof regularly. as long as youre doing overhand pullups you'll be helping yourself get in shape for what's to come. having just come out of Fort Benning recently i'll tell you that you'll look bad if you can't do at least five, then try working on swinging your legs up and around the bar.
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