If you want to live off the grid, you'll have to fell trees. You may be building a cabin, or processing firewood. As far as science knows, trees possess no central nervous system. Save the hippie guilt for butchering animals.

Preparation

Unsurprisingly, felling timber is dangerous work.

First, evaluate your safety gear. You should literally be covered head to toe — hard hat, eye protection, steel-toe boots, and, if you're working with a chainsaw, clothing made of fibers designed to bind the chain before it reaches your skin.

Laying the groundwork for the actual cut is a fairly complex process. Of course, all possibilities of mishap must be taken into account before the tree starts to tip over, since thanks to the laws of physics one is ill-equipped to stop a tree which has gathered even a small bit of momentum. Take stock of your surroundings, noting the location of powerlines and other trees. Removing a tree that is leaning against another is very dangerous and difficult work, and dropping a tree on phone lines will start a fire. Also take time to think about whether any branches on your candidate tree are rotten or prone to dropping, since the momentum of the tree's shaft swaying on a hinge will likely cause them to break off mid-flight.

A good way to break a two-by-four is to slam it against a sharp edge; a good way to break a felling tree is to drop it on uneven ground.

Few trees stand straight. Things fall in the direction they lean.

To keep the tree from swiveling around and dropping at a bad angle, tie a safety line to it directed at the angle of desired fall. Naturally, you want to attach the business end of your line to a point high up on the tree, utilizing the same principle that places our doorknobs at the outside edge of the door.

Plan an escape route 45° from the sides and back of the tree — think eight and four o'clock. Never stay directly behind a falling tree, lest it slip off while falling and kill you.

Cutting

A tree is felled with two cuts, each on opposite sides of the trunk: the undercut, and the felling cut.

The undercut is underneath the felling cut. The felling cut is what fells the tree. The undercut comes first. The two cuts meet roughly at the center of the tree's girth.

Make the undercut at a height ¼ the diameter of the tree.

        |.........../ ---Direction of fall---->
        |........../
        |........./
        |......../
         \....../
          \..../
           \../
 Felling cut\/<-----Bridge
        |.../_____Undercut
        |...........|
        |...........|
        |...........|
        |...........|
        |...........|
    ___/....Trunk....\___

If you're using an ax, make sure the blade hits the surface of the tree at 45°. Hit it straight on, and you won't cause any damage; make the angle of contact too large and the axe head will deflect.

If you position your undercut and felling cut correctly, the difference in height will act as a hinge that braces the toppling shaft of the tree against the trunk. Naturally, when you drop the tree, push it from the side with the higher cut so that it doesn't slip off and crush your pelvis.

And only a fool pushes a tree over with bare hands.

As seasons pass, trees add layers of flesh. Accumulated flesh forms grain. In a live tree, grain runs vertically. Wood splits with grain — smash the end of a two-by-four sometime. Push a tree over and, occasionally, it'll snap right up the middle, and a rubber-band effect will spring one of the halves in the direction of the higher cut: precisely where you're standing.

To prevent your own death, push the tree over with a long pole at a healthy distance. Expolit the principles of leverage by pressing as high on the tree as you can reach. Crack— the tree starts to topple. The bridge you cut prevents it from slipping over its own trunk. Thanks to the angle of your cuts, it drops at exactly the angle you intended with a satisfying crash and rattle-hiss of thousands of leaves.

Assuming there's no wind.

Did you remember your safety line?

Trees are heavy — in many cases, heavier than roofs, heavier than cars, and in even more cases heavier than skulls and ribcages and soft lungs. Remember your safety line.

Preparing your felled tree

Trees are heavy enough without the friction of branches and thousands of leaves dragging on the ground. Before moving your tree, you'll have to trim the branches.

Stand alongside the felled tree on the opposite side from the branches you're cutting. Use the 45° rule again to maximize the damage of each cut. Sweep downward with the axe, in the direction the branch is growing. Like shaving with the grain.

If you've felled a tree that's too large to move even with the branches removed, chop it into a series of logs. Stand on top of the tree and strike between your legs. Again, use the 45° rule, pointing the ax blade toward the end of the tree furthest away from your position. Cut a wedge from one side tapering to the center of the girth; then turn around and do the same on the opposite side.

Splitting logs can be tedious work.

If you've got a short log, you'll want to split from the top. Place the log in a chopping block (a beefy Y-shaped section of tree should work nicely). Swing your ax straight down into the center from the top, twisting the head slightly when it makes contact to pry the wood apart. Strike as many times as needed. It's not like television — you've got to be hardcore to knock a block in half with one stroke.

If you're splitting a longer log, the above method is useless. Instead, you'll have to use wedges. As before, make a starting cut at the end, straight down, ax head twisted slightly on contact. Then pound wedges into the cut to hold the sides apart. Next, cut along the side of the tree along the top cut, hammering wedges in as you go. Eventually, it should peel apart like string cheese.

You may like your logs de-barked — this is likely the case if you're planning on making good lumber. If so, bark is best removed in the Spring, when the tree is full of sap. Cut a straight line for the length of the log and pry the bark off using a wooden spud — a length of wood some 2' long with a wedged end. Try this in the Summer and you'll be chipping the bark off one splinter at a time.

Stumps are wasted space. You'll want to pull them.

Usually, getting stumps out of the ground is very dangerous work. It involves applying tension to ropes nearly to breaking point. Imagine a rubber band breaking to the power of ten thousand.

A homemade stump winch-puller is a quick and dirty solution to unsightly stumps. Between the stump and a solid anchor — preferably another large tree — drive a thick stake sans bark perpendicular to the ground. Next, cut the exposed stump roots and, if it's a large stump, dig a pit around it. Also dig a two-foot deep downward-tapering hole around the stake you've driven.

Next, attach a very strong line to a strong root on the stump's far side. Tie another pole, 6' to 8' long, perpendicular to the driven stake at chest height to act as a lever. Finally, attach the line to your anchor.

With one person at each end of the lever pole, twist to increase the tension on the line. For fuck's sweet sake, go slow. Move a little at a time, stopping frequently. If that sucker pops you'll be out of commission for a very long while. When you hear the stump start to crack, stop — the tension of the line might finish the job for you. Otherwise, keep going. Sloooow.

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Sources

Tawrell, Paul. "Camping & Wilderness Survival: The Ultimate Outdoors Book." Tawrell: Ontario. 1996.

About.com
http://http://forestry.about.com/od/chainsaws/ss/fell_tree_2.htm
http://http://forestry.about.com/od/chainsaws/ss/fell_tree_7.htm

Grounds Maintenance Magazine
http://grouds-mag.com.mag/ground_maintenance_fell_tree_opennotchandbore/index.html

Essortment
http://mo.essortment.com/howtucutdown_rctq/htm

The Traditional Handbook
http://www.inquiry.net/outdoor/skills/b-p/felling_tree.htm

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Note: I have done this once, so consider my advice null and void. Do not listen to me. Seriously, you will die. Cheers!