This song was played by Pete Seeger on his concert in Prague, (then) Czechoslovakia in 1964. The live record of this concert has been published as a disc 1 of this double CD: cduniverse.com/search/xx/music...+1964.htm You can find the song as a track #18.
Some background information of Seeger's concert in Prague (beware of entering the forbidden territory of politics) is at genedeitch.com/7.html . Speaking about 5-string banjo - Seeger's tour was definitely one of the main triggers of 5-string banjo and bluegrass popularity in the Czechoslovakia / Czech Republic. Reed more e.g. at en.wikipedia.org/wiki/Czech_bluegrass . Before that concert, the word "banjo" meant tenor banjo in our part of the world.
With the exception of the very first note the song is similar to the Czech children folk song Holka modrooka (translates as Blue Eyed Girl). This fact and also the link to Uncle Dave Macon is mentioned in the Wayne Erbsen book Old-Time Fiddle for the Complete Ignoramus! books.google.cz/books?id=RFCD1L_XMO0C (look at page 38). Simple banjo tab in double C tuning is at page 28 of Erbsen’s Clawhammer Banjo For The Complete Ignoramus! books.google.cz/books?id=mPB2Rkh0UvsC
For singers, here is the karaoke aid for Holka modrooka: youtube.com/watch?v=vTOUuhAxR8k It is in the key of C and melodic clawhammer players might be inspired by the solo. No, it is not common to harmonise this song with Cdim (instead of F) in the second half of first measure and with the I-VI-II-V-I turnaround (instead of plain C – G – C) at the end of the verse.
I have this, with Pete Seeger telling a story and making sound effects on his banjo, on Birds, Beasts, Bugs and Bigger Fishes, a ten inch Folkways recording that I got as a child when it first came out, and which autographed copy I still have.
If I would have to pinpoint one song that most led me into playing the banjo, this would be it! I listened to it over and over as a child, and play it in a variety of tunings and settings (though I haven't played it for years).
I lifted the general idea of the inner story with banjo and vocal sound effects in Seeger's version of this song for my version of "The Fox" that appears in the music section of my page.
The Cumberland Mountain Deer Chase is not the only old time tune with Eastern European roots. I have a recording of "Flop Eared Mule" by a Ukrainian trio, recorded in NYC in the 1920s (There is at least one note different but that note is not available on a western fretted instrument!).
That tune obviously didn't come to the US in the 1920s, but probably arrived with the Eastern European miners brought in during the 19th century. By importing small groups from several eastern populations, the mine owners kept the workings from unionizing - they couldn't communicate with each other. Before anyone claims this is political, it is simply history. The reason why so many Welsh tunes sound Eastern instead of Celtic, is because the English mine owners were doing the same thing starting in the mid 18th century.