I looked it up, it's pretty cool! I'd never heard of that before.
I have no idea how you would keep it from falling over... The only thing I could think of was to take a long 4x4 piece of wood, drill a hole in the center that's pretty deep, and the exact diameter of your tree trunk, and then bury the piece of wood so that it's flush with the ground, or just a bit under. Then put the 'trunk' in the hole. It might be more sturdy than just planting the thing right in the ground. You could mow the lawn around the piece, and just pass the whipper snipper around the edges where you don't reach with the mower.
Just a suggestion :)