The earliest reference I've found to the usual American form of the song is this:

The Honey Jar: A Receptacle for Literary Preserves (Columbus, O.) Vol. 9, 1907, p. 102:

"....where high-priced birds sing the latest songs, and the green grass grows all around, all around."

When I was in grade school in NYC (ca1957), we sang this in chorus.

As I recall, it went:

Now in the woods there was a tree,
The finest tree you ever did see:
The tree in the wood
And the green grass grew all around, all around,
And the green grass grew all around.

Now on the tree there was a limb,
The finest limb you ever did see:
The limb on the tree and the tree in the woods,
And the green grass grew all around, all around,
And the green grass grew all around.

[Similarly:]

...branch...
...twig...
...nest...
...egg...
...bird...

It would have been "circular" had it started out with a bird on the tree, but I don't think it did - at least not as far as I can recall.

The tune differed from "The Rattling Bog." The rhythmical "green grass"refrain was almost identical to the concluding two lines of the refrain of Stephen Foster's "The Glendy Burke" - though we didn't know it at the time.