Alas, I don't know the tune to Kitty Alone. However, the structure of the verses reminds me of a childrens' song I learned, which we called The Frog In The Well. IIRC, the first verse went like this:

There was a frog lived in the well
Heigh Ho, said Rollie.
There was a frog lived in the well
And a merry mouse in the dell
With a rolly polly, cabbage and spinach
And Hi, Sir Anthony Rollie.

Only you who can read notes and perhaps know this one can tell if the two are the same or share the same tune. I would be curious to know if they do.