A quick check of the DT doesn't turn up "Tit Willow", so I'm posting it now.
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TIT WILLOW
(W S Gilbert & A Sullivan)

On a tree by a river a little tom-tit
Sang "Willow, titwillow, titwillow!"
And I said to him, "Dicky-bird, why do you sit
Singing 'Willow, titwillow, titwillow'?"
"Is it weakness of intellect, birdie?" I cried,
"Or a rather tough worm in your little inside?"
With a shake of his poor little head, he replied,
"Oh, willow, titwillow, titwillow!"

He slapped at his chest, as he sat on that bough,
Singing "Willow, titwillow, titwillow!"
And a cold perspiration bespangled his brow,
Oh, willow, titwillow, titwillow!
He sobbed and he sighed, and a gurgle he gave,
Then he plunged himself into the billowy wave,
And an echo arose from the suicide's grave--
"Oh, willow, titwillow, titwillow!"

Now I feel just as sure as I'm sure that my name
Isn't Willow, titwillow, titwillow,
That 'twas blighted affection that made him exclaim
"Oh, willow, titwillow, titwillow!"
And if you remain callous and obdurate, I
Shall perish as he did, and you will know why,
Though I probably shall not exclaim as I die,
"Oh, willow, titwillow, titwillow!"

Notes: From "The Mikado" sung by Ko-Ko to Katisha.
There are similarities here to the song sung by Desdemona in Othello Act IV Scene III, which also appears to be an early variant of "All Around My Hat", in that it mentions the wearing of a green willow garland.
NP