It did in fact come from one of those and the song is actually called The Walnut Tree. I give the words and music below. Thanks again to those who looked for it.

Mick

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THE WALNUT TREE
(Roger Fiske)

Bill Palmer, the farmer, was not a clever man.
He planted a walnut; that's how it all began.
He chose to put it close to the side of his house;
It wasn't no bigger than any small mouse.
It wasn't no bigger than any small mouse.

The walnut grew tall but it hadn't room to spread;
A big tree should not be too near a house, its said.
His wife Jill said 'Look Bill, the thing about that tree,
It blocks up the winder so no one can see.
It blocks up the winder so no one can see.'

To calm her Bill Palmer said, 'Never mind, my dear;
I'll climb up and cut off the biggest branch what's there'.
He had a long ladder, and with his large saw
He climbed up and sat there; Jill stood by the door.
He climbed up and sat there; Jill stood by the door.

But while he was sawing, his wife said, 'Look at you,
You're sitting the wrong side of where you're sawing through.
Oh Billy, you silly, you'll come such a bump.'
As she spoke the bough broke, and Bill fell down 'clump'!
As she spoke the bough broke, and Bill fell down 'clump'!



Source: The Oxford School Music Books, Junior Book 1 by Roger Fiske and J.P.B. Dobbs, OUP, 1954

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X:1
T:The Walnut Tree
C:Roger Fiske
L:1/4
M:3/4
K:G
G | A G G | B G G | c/G/ G G | d2
w: Bill Pal- mer, the far- mer, was not a clev- er man.
d | e c A | d B ^G | c/B/ A =G | A2
w: He plant- ed a wal- nut; that's how it all be-gan.
G | A G G | B G G | c G G | d2
w: He chose to put~it close to the side of his house;
|: d | c d d | B d d | A d d | G2 :|
w: It was- n't no big- ger than an- y small mouse.