Once upon a time all trees were cut down -- felled -- with handsaws. Even in today's modern world, many, such as the Amish and Mennonites, still use handsaws to cut, well, everything. It may prove a little more labour intensive to use a handsaw, but it tends to be safer than a chain saw. Keep in mind that trees can give way suddenly and proper procedures -- along with common sense -- will help protect you.
Stand back from the tree you want to cut and look at it closely. Note where heavy or precarious branches and limbs are, which could prove difficult or hazardous when felling the tree. Identify which direction the tree seems to lean, or even which side of the tree has the most branches. This will prove the "heaviest" side of the tree, and the direction in which the tree will want to fall; consider this side the "front" of the tree, for cutting purposes.
Trim branches and limbs that will make cutting the tree down easier, or could prove dangerous during your work. Use your handsaw and make a cut on one side of the branch, cutting about a third of the way through. Switch to the other side and continue cutting until the branch falls off. Use a pole saw to reach limbs over your head; the procedure is the same, the saw is just at the end of a long stick or handle.
Cut a horizontal cut low on the "front" of the tree, going through about 40 per cent of the tree's thickness. Work the saw back and forth to create the cut, applying firm and steady pressure. Make another cut just above the first, coming down at an angle so the two cuts meet like a piece of pie removed from the rest. Avoid cutting through more than 40 per cent of the tree or it may fall prematurely.
Move to the back of the tree and locate a spot on the tree about 2 or 3 inches above the tip of the pie cut. Make a "back cut" at this point, parallel to the pie cut and to the ground as well. Make sure you are indeed above the first cut before cutting, or you will be unable to control the direction of the fall.
Continue cutting through the tree trunk until the tree begins to fall. This will not take cutting through the entire trunk, or even half way through; as the joint where the middle of the tree remains, between the back cut and pie cut, narrows, the remaining wedge will weaken and give way. Calmly step away from the tree and wait for it to fall; this should prove to be away from you if you have cut it correctly.
Very small trees, 1 to 2 inches thick for instance, can usually be pushed over once you cut through part of the tree trunk. Never push on a large, thick tree.
Be aware that even if the tree falls away from you, the stump can bounce and fly back up at you. Plan where you will stand before you begin cutting the tree, and do not panic when the tree begins to topple.