Who wrote the poem beginning one bright day in the middle of the night two deadboys got up to fight?

It's an old folk tune. Some people say that the author is unknown. Some give the credit to "Ed Cray" of British Columbia. Others say Shel Silverstein.

Hmmm. Seems like a lot of people are given the credit for it because they have used it in their writings long after the earliest records of the poem.

Given the age of the poem, it more likely is an old folk tune and the author probably is unknown. Those are the earliest references to the poem, so it definitely pre-dated Shel Silverstein. There is no reason, though, that he or any other author couldn't use the poem in their writings without giving an author credit for it.