Funny but it seems that "Princess Pat" mentioned above is not
discussed on mudcat.org. I did find one text:

The Mudcat Café TM
Thread #2795   Message #1875270
Posted By: GUEST,Random college student
03-Nov-06 - 07:26 AM
Subject: RE: Heigh Ho, Heigh Ho, I Bit the Teacher's Toe!

A ricky bamboo (a ricky bamboo)
Now what is that? (now what is that?)
It's something made (it's something made)
By the Princess Pat (by the Princess Pat)
It's red and go-old (It's red and go-old)
And purple, too (and purple, too)
That's why it's CALLED! (that's why it's CALLED!)
A ricky bamboo (a ricky bamboo)
Now Captain Jack (now Captain Jack)
Had a mighty fine crew (had a mighty fine crew)
He sailed across (he sailed across)
The ocean, too (the ocean, too)
But his ship did sink (but his ship did sink)
And yours will, too (and yours will, too)
If you don't HAVE! (if you don't HAVE!)
A ricky bamboo (a ricky bamboo)
[repeats from the top]

Text retrieved from here: https://mudcat.org/detail_pf.cfm?messages__Message_ID=1875270

Supposedly from WWI.