The song "Deep in the Meadow" is a lullaby Katniss used to sing to Prim, and she sings it again when Rue is dying to comfort Rue. This lullaby would have been common among District 12 children and the lyrics and melody would be warm and relaxing, showing that even though poverty and tragedy is rampant among District 12, they have ways of retaining their humanity and coping.
"The Hanging Tree" is a song Katniss learned as a child, but she does not realize the meaning of the song until she is older. She sings it in the books/movies during a propo. The effect of this haunting song is that it unites the people of the districts against the Capitol because they can relate to viewing death/suicide as an alternative to the unfairness of living in Panem under the Capitol's rule.
The music in Appalachia is soulful and country sounding. Bluegrass and banjos are popular in Appalachia. Some of the ballads have similar subject matter to the ballads in the Hunger Games. For instance, the song about murderer John Brown being hanged is similar to the hanging tree, which is also about the death of a criminal and its effect on the town.
A difference between Appalachian songs and District 12 songs would be the instruments. Appalachian ballads were often played on guitars or banjos, whereas the people of the Seam did not have the money for instruments and seemed to rely more on their voices. Both the music of the Hunger Games and the music of Appalachia had lyrics about their geography. Both peoples love their forests and mountains. The people of the Seam sing about valleys and forests, and the people of Appalachia sing about their mountains.