In my opinion, no one has provided an exactly correct answer. Now I'm not argueing with anyone's input, I just think you have all been missing the point of that 1/3rd diameter face cut.
1/3 because there’s no need to cut extra wood. The face takes 2 cuts. The back cut takes 1. 1/3 will generally give you 80% of the diameter . Plenty of hinge.
I agree. It's that "plenty of hinge" concept that is important.
The best argument for using wedge and 1/3rd face cut is because (as has been pointed out above), you can be wrong, and then the tree will "set back" on the chain. At least in theory, you have chosen the face cut direction because of the lay of the land and the direction you need to have it fall. After that choice is made, you need to plan for how to best get it to go that direction, and also to plan for potential failure.
That fairly shallow face cut is more about the tree potentially going the wrong way than it is about getting it to go the way you want. By widening the length of the lever formed between the wedge and the hinge wood, you increase the effectiveness of the lifting force applied by the wedge, and you seriously reduce the risk of hinge failure, should the tree resist going the direction you wish it to go. Keeping that hinge wood as wide as practical without reducing the depth of the backcut puts you at 1/3rd diameter. Roughly speaking, that is.
Consider, if you will, that you might just choose to drop a 3 foot diameter tree that is 100 feet tall. It's straight and true, with no breeze. Theoretically, you should make a one foot deep face cut, then start a back cut. This will leave you with a theoretical lever on tipping the tree of 50:1. That's a 100 foot tall tree, with you working on lifting a 2' wide lever at the base with a wedge. Ok?
Now let's toss caution to the wind. Instead pf wedging that tree over with the recommended procedure, we'll make a 2 foot deep face cut, undercutting the center of gravity. Then a quick back cut, and we needn't wedge anything at all, right?
This will work, but it is fundamentally unsafe. Let's say that just as you start your back cut with only 1 foot of trunk diameter remaining to cut, a slight headwind picks up, and your tree now wants to go forcibly the wrong direction. Even if you make a bore cut to allow insertion of a wedge to force the tree against the breeze, your leverage at the base is now less than 1 foot wide, against a 100 foot tall tree. So you have cut your leverage in half, and you have doubled the lifting force applied to the hinge wood keeping the tree on the stump.
OOPS! BAD Plan!
(panic ensues) Quick...Pull that tree with the rope.
Oh crap. I never set a rope either, and the breeze is picking up.
<cracking noise!> NO! NOooo
ooo.
...As your easy-to-cut-down tree flops backwards in the breeze onto whatever you were protecting.
If the tree were perfectly balanced you would have a point. Few are. An error of judgement leaves the feller without enough wood to use wedges.
Yes, but you didn't explain how that was a problem.