No relation at all to House Carpenter, so far as can be told. The basic scenario is common in French tradition, but develops in various ways depending on the song-group involved. Most have been discussed here before.

It's my (purely personal) feeling that the Canadian Maid on the Shore is not really related to the Broomfield Hill group at all, as received wisdom would have it; nor does it derive from some putative Irish ballad as has also been suggested. I don't see how it can be other than an English-language redaction of a French song.