The F-51D was used in Korea rather than any of the somewhat different lightweight Mustangs (H) or morever the apparently more suitable F-47D or N because not only did more F-51D's remain in total USAF inventory (though the difference between total F-47 and F-51 in June 1950 wasn't as great as is sometimes portrayed) but those in ZI were concentrated on the West Coast, and moreover the FEAF still had a small number of F-51D's in inventory in Japan (though not distributed to units).
My father commande the 35th FBW equipped with 51D's out of Johnson AFB from 1949 to early 1950.
Those were the a/c sent to Korea in late Jun for the ROKAF but which flew some missions w US pilots. Also it had spares stockpiles and remaining maintenance and pilot experience with the F-51D, not with the H nor the F-47. The H would have offered little advantage in Korean conditions anyway, D being adequate in the handful of combats pitting F-51's against NK prop a/c and any prop fighter was heavily outperformed by the MiG-15 later on. But FEAF later regretted the decision of 51 v 47 as F-51 losses soared especially in 1951 v more heavily AA armed Communist field armies by that time. But, the USAF wasn't willing to support operation of multiple prop fighter types in Korea and (commanding general) Stratemeyer was told to forget it when he broached the idea of adding the F-47 later on (see "Stratemeyer Diary").
Joe