" A Coolabah tree is a species of tree. It's very shady."

...from "Forest Trees of Australia" by 8 renowned botanists..."Eucalyptus microtheca has a wide geographical range and an associated variation in size from a tree 15-20 m in height, with a moderately well-formed bole one-quarter to one-third of the height and up to 1 m dbh, to a poorly formed, small tree with an open, straggly crown and a very short trunk."

the photographs they show with it sure don't look to ME like a tree renowned for it's shade!~

I suspect EITHER that using a Coolibah in the song (rather that a Kurrajong, which IS shady) is part of an in-joke, or simply that is was a well-known tree whose name fit the rhythm scheme. (and I suppose that if a Coolibah is all that was around near that billibong, that's what he'd use....still...*grin*)