What children's song is also known as "William Trimmytoes"?

The title of the book/movie "One Flew Over the Cuckoo's Nest" comes from a children's folk rhyme. It is a counting rhyme; akin to One potato, two potato or Dip, dip, dip my little ship. Like most oral folk verses, changes have happened over time and the origins are difficult to determine. There are two distinct versions that I have found, and not knowing which is the original, I've decided to list them both.

Chief Broom mentioned this in the book; he comments that it was a game he would play with his grandmother when he was a child:

Tingle, tingle, tremble toes
She's a good fisherman
Catches hens
Puts 'em in pens
Some lay eggs.
Some lay none.
Wire, blier, limer lock
Three geese inna flock
One flew east,
One flew west,
One flew over the cuckoo's nest
O-U-T spells out
Goose swoops down and plucks you out

This is variant of the William Matrimmatoe chant:

William Matrimmatoe
He's a good fisherman.
He catches hens,
Puts them in pens.
Some lay eggs.
Some lay none.
William Matrimmatoe
He's a good fisherman.
Wire, briar, limber, lock.
Three geese in a flock.
One flew east.
One flew west.
One flew over the cuckoo's nest.
Wire, briar, limber, lock.
Out goes you, old dirty dish rag, you.

The William Matrimmatoe chant has several alternatives. "William Matrimmatoe" is sometimes replaced with "William, William Tremble Toe," "William a Trimmy Toe" or "William Trimble Trow Tran." Also, it sometimes ends with "But O-U-T spells out. So begone You dirty trout. You."

This rhyme is commonly credited with the origin of the title:

Vintery, mintery, cutery, corn,
Apple seed and apple thorn;
Wire, briar, limber lock,
Three geese in a flock.
One flew east,
And one flew west,
And one flew over the cuckoo's nest.

It is a variant of this Mother Goose rhyme that was printed in 1814:

Intery, Mintery, cutery corn,
Apple seed and briar thorn:
Wire, briar, limber lock,
Five Geese in a flock,
Sit and sing by a spring,
O-U-T and in again.

These two verses have some similarities and probably have common roots. However, I cannot find any information that links to the two. If anyone out there has any more insight, please let me know or note it yourself.

My source:
http://www.everything2.com/index.pl?node=One%20Flew%20Over%20the%20Cuckoo's%20Nest

william william trimmytoe
he's a good waterman
waters his hens
puts em in pens
some lay eggs
some none
wire briar limber lock
three geese in a flock
one flew east
one flew west
one flew over the cuckoo's nest
mouse ran around
clock fell down
o-u-t spells out
you old krout

farley farley finklestien
he's a good waterman
waters his swine
some for bacon
some for ham
some for porkchops in a pan

I also played this game with my grandma when I was a little boy. This is all I can remember of it. She was from south central Kentucky.

I'm not sure of the song, but my commentary is on an old chant from back in the early to mid 1940's in central Kentucky around Frankfort. It was used to choose game participants, decide who was "it" in several games, or otherwise serve the function of selection by elimination.

All the players would stand in a circle with one or both fists extended toward the inside of the circle, one of the older players would stand in the middle of the circle and begin the chant, counting one fist with each accented syllable, starting with himself.

Wire briar, limber lock (4 counted)
Three Geese in a Flock (4 counted)
one flew east and one flew west (4 counted)
One flew over the Cuckoo's Nest (4 counted)
O-U-T, and out goes he (7 counted)
Out into the deep blue sea (7 counted)

The last counted was out. Then it would start over with the rest, until it was down to one. It is interesting to note that this was a 30-count elimination round.