I presume ANICE refers to the 4-vol. set published and copyright by Princeton Univ. Press, Bertrand Harris Bronson, "The Traditional Tunes of the Child Ballads," 1959ff, and later the abridgement, "The Singing Tradition of Child's Popular Ballads," copyright 1976, also by Princeton Univ. Press.   

The previous post by Malcolm Douglas succinctly states that not all tunes included in Bronson are in the public domain.
Looking at just one of the ballads, "The Three Ravens," includes a tune sung by William Nelson, a version published by Creighton and Senior, 1950. That volume and its contents are under copyright. separately from the printing in Bronson, and there may be a copyright by Nelson.