when trying to cut down a tree, your not looking to blow one big hole in it, your looking to weaken the structure of it so that it falls.
its the same deal with taking down a building with explosives. you dont try to blow every last bit of the building up. what you do is place explosives onto the load carrying parts, damage those until they cant do their job, and watch the bulding fall under its own weight.
i think a good houserule is that when a material is below half its structure, it will fail structuraly. so to slash a tire you only need to do is about 1 point of damage.
allso, armor in SR4 isnt straight up damage reduction, its rolled. so you may not do any damage (damn hard patch of wood or something) or you may be able to cut the tree down in one swing (nice work). but if you trade the dice in, both 4 armor and 7 armor will result in -1 overall damage, so the axe will do more damage then the club...
btw, 7? isnt 4 doubled, 8? so make that damage reduction for the axe -2, still thats 5 damage(axe) vs 3 damage(club) if the gm use the dice trade rule

basicly, the barrier rules in SR4 is buildt to do two things. first is to allow you to use it as extra armor (shooting through barriers), the second it to allow you to make your own doors (destroying barriers).
neither of those are realy appliable to cuting down a tree, or slashing a tire...