Quick correction to the above:

G.B. Grayson recorded the song as "Rose Conley."

It was the Mainers who retitled it "Down in the Willow Garden."

Such retitling was a frequent practice in the recording industry of the time, to conceal the fact that one band was covering the song of, or in any way using the repertoire of another.

So there's nothing magic about "Down in a willow garden," it's merely the first line of a song that until then apparently had never been titled that way.

Bob