An extreme case
By now you are probably completely sick of those little boxes I've been drawing on the pictures, so I won't do it here. Just notice that I needed 2 mages to block because that '1' tree in the picture is actually the size of a farm. This is a narrow path I chopped on 3v3.pud towards an enemy. Once this peon is on the other side, the enemy is in some real trouble. I can easily protect any buildings with my mages. By using my newly made mage tower on the other side of the trees, I can pump mages, and hit their peon line, or whatever else I feel like.
As an expansion
By chopping 1 tree, I set myself up for an easy hop inside the middle are of this map. For those of you who have never played ant trails, there are 2 mines you can build a hall in between. By hopping in like this, I keep the center area walled in by trees. A very sneaky expansion.
This is helpful when playing Arctic Circle, from the bottom right position. Unfortunately the map was designed poorly, allowing both top starting positions, to get through to the middle by only using 1 sapper. Both bottom spots require 2. Using this hop you can get to the middle area by only using 1 sapper. This exact hop has actually won me a game.
By just chopping 1 tree, you can create so many new hopping opportunities that I could never discover them all. On the map nowhere to run, try doing some 1 chop, hops. There are a bunch of fast ones that you can do. Even on maps like Garden of War, there are some pretty neat 1 chop, hop, spots (say that 3 times fast).