What are the complete lyrics to the popular song that contains "I
don't know why she swallowed the fly, perphaps she'll die." It also
contains "she swallowed the X to catch the Y, swallowed the Y to catch
the Z," etc.

Hello.

Peter, Paul and Mary have the lyrics for a common version (copyrighted in 1952):
http://www.peterpaulandmary.com/music/17-07.htm


The popular version recorded by Burl Ives is slightly different,
featuring the last line, "She died, of course" instead of the more
upbeat "She's alive and well, of course!"
See:
http://www.traditionalmusic.co.uk/song-midis/Swallowed_a_Fly.htm
Also see:
http://www.niehs.nih.gov/kids/lyrics/oldlady.htm

Here's a bit on the history of the song:
"Alan Mills, ebullient Canadian singer came across these these verses
in a magazine, changed them a bit, added a few more, and added a tune.
Countless, future generations of children will enjoy the result."
http://www.mudcat.org/thread.cfm?threadid=7256

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search strategy:
"burl ives" "she swallowed the cat to"
"alan mills" "she swallowed the cat to"

I hope this helps.