I just started playing The Lorax, and I must be missing something because I can't figure out how the Onceler would go about winning.  Can someone explain how this works?

I think Onceler has to take all the trees.  These are all capitals, and the Lorax starts with a few, so take all the trees and the Lorax is out. 

It's harder as the lorax, because you have to take all the trees and hold them until the beginning of your next turn to kick in your factory continent that will eliminate your opponent.

Check out the discussion on the tournament also:

http://www.wargear.net/tournaments/view/806

It would be a lot more clear if the history worked, but all the games I have checked, the history is broken.

Well, let me be more specific.  How can the Onceler take out all the trees when he only has 3 attacks per turn?  The Lorax has auto-capturing factories on some or all of the foliage, depending on what else he owns.  Once you own the spade, that alone would be 5 territories per turn the Onceler would have to kill in order to win?

The Once-ler has some negative factory bonuses on the trees as well (all listed in the description) that help him slow down the Lorax's growth enough that the chopping can finish him off:

• The Chop-o-tree 3000 knocks each layer of foliage back by -1 each turn.
• The Toxic Sludge puts a -1 factory bonus on the water bucket each turn.
• The smog puts a -1 factory bonus on the pedestal (old stump) each turn.
• Controlling both the Chop-o-tree 3000 and the Axe put a -1 factory bonus on the Lorax each turn.

What really keeps the Lorax going is collecting bonuses from the Bar-ba-loot and the Stump, which he can then fortify into the trees through the water bucket. Anything the Once-ler can do to slow that additional growth will go a long way toward whittling away the Lorax's impressive regenerative ability. Note that the Once-ler has options to attack the stump and the bar-ba-loot directly, though you give up an attack on the trees to do so, of course.

but for the Once-ler to win, on their turn all the levels of foilage except 3 would have to have just one unit on them. that seems very difficult to achieve, with the water bucket, spade, and the bonuses from the foilage themselves. It seems to me that this board is rather heavily slated towards the lorax.

It also seems clearly neccessary that the once-ler gets the chop-o-tree 3000 quickly, which is very hard since it starts with 7 troops and you only get 3 attacks. By that time the lorax could have the spade and water bucket and all the foilage  occupied, many with over 5 units.

The Onceler starts with 10 units, and an additional 10 to place, so he can take the chop-o-tree first turn if he wants.

The Chop-o-tree 3000 knocks each layer of foliage back by -1 each turn.

I missed this somehow, that does seem to even things up a bit.