Ignoring the rules for a second, if you think it out logically, one could only cleave into a new target if the initial target were somehow removed from your weapon's path. If you go by that logic, anything that forces your opponent out of your path could theoretically allow a cleave (including things like the Brutal Surge weapon enhancement).
Now I can't claim to have any insight into the designer's intentions, but the flavor seems to be that if you swing your weapon so hard that it clears an obstacle from the weapon's path, then you may continue with the original momentum of the swing and hit an additional target. That being said, I really don't think a standard trip attempt would qualify, as you aren't hacking at your opponent with a ton of force, so much as you're simply trying to push them off balance. Knock-down, however, would seem to qualify as you are hitting them normally, and you just happen to know how to use the force of that hit to put them on the ground.
I'm still not personally sure which side of the issue that I'd come down on if I were DM'ing, but a case could clearly be made for either side.