If Cora (Barbara Stanwyck) had lice (the reason given for her cropped hair), her son would have had lice, too, due to all the direct contact between them. Probably, half the tribe (at least) would have contracted lice, and yet she was the only one with the cropped hair.
Head lice are very specific to type. Caucasian lice, Asian lice, African lice, even animal lice (cats, cows, dogs and pigs) all have different characteristics to grip the hair of their victim.
Even though the story takes place some time soon after the Civil War, Barbara Stanwyck's hair style is strictly a 1957 perm.