Any organic matter, if left in a pile (or a layer on the ground, like mulch) will eventually decompose. That said, there are ways to make it work better and more efficiently.

When you put things in the pile, balance browns and greens. Ideally, if you mix up a big batch at once (say, leaves and grass clippings) that's well balanced, it will make a pile of compost without doing anything else to it. Most of the time you can't do that - you have small amounts of kitchen scraps, yard and garden trimmings, etc. Keep a supply of browns nearby - leaves, shredded paper, straw, sawdust, cardboard, etc. When you add greens, layer browns on top to cover. The layer cake will become compost down underneath after a few months.

More turning allows mixing, aerating and moisture adjustment, which makes compost faster, but you'll still get compost the lazy way.

I have a pile of shredded trees. So could I put some down, and everytime I throw out scraps, just add some more on top?

Work "better"? If one were to just pile everything organic that one can find in a heap or in a bin and wait enough time, it will decompose and form compost. Without turning. Without balancing. Without calculations. Without inventory in reserve. Without adjustments. All you need is time and the realization that make-work in a garden is for "efficient" people.

I do the pile and forget method. When I want compost I break down the oldest bin. This spring I broke down 4 bins and covered my veggie garden in about 6 inches of compost/mulch.

http://voices.yahoo.com/building-cheap-wire-compost-bin-2612834.html

You do have to keep them moist, and it takes more time and space (maybe) than the turning method. But for sheer laziness it's awesome.

Here is a link that might be useful: Cheap wire no-turn bins

"Is there a way to do your own composting without having to turn it?"
Certainly. That is how Ma Nature makes compost. Just pile up the material and wait. No need to even pay attention to the moisture level. That method, however, can take years before you get anything usable.

"I have a pile of shredded trees. So could I put some down, and everytime I throw out scraps, just add some more on top?"

That is perfect. Turning helps speed things up but you don't have to. Sooner or later it will be compost.

I am gathering that this will work, it just may take a pretty long time to work. So would it be better to use a DIY tumbler that wouldn't have the contact with the soil, or use the pile/bin method, but not turn it that often?

Many people I know that bought tumblers did not have much success with them and thought them a large waste of their money. Tumblers will require turning, do not have much volume, and often are just an eyesore.
A pile, or a containment bin, will be a better choice for you for what you plan to do.

IALBTC.

I agree with kimmsr about the tumblers. The problem with them is they don't hold a lot of volume, so you've put a lot of effort in for a small amount of compost. And the ones in my stores are very expensive. For the price, I could have bought yards of finished compost from the quarry.

I bought one of those plastic bins that open on the top and have a "door" on the bottom that you were supposed to shovel your finished compost out of. That was a colossal waste of money. The thing was a snail and bug magnet. Because you can't turn compost in the bin, the outside edges never broke down, thus it was very difficult to get anything out of the "door". In the end, it would have been easier to just have an open pile that I could access better.

For my method to work, you need a pile at least 3x3x3 ... and you can't get a tumbler that big.

The outer edges and the top layer don't break down much - getting the compost out is like getting the filling out of a cream-filled cupcake - there will be a 6-inch crust around the finished compost.

The "crust" - the edges and top layers - get tossed into another bin to keep decomposing. So I never get 100% yield on any bin but I have way less work.

Usually a "fresh bin", one that was filled with all new material, will give a lower yield than a bin that's got layers of things that have been in a bin before.

I never turn, like lazygardens. I can empty at least two compost bins onto the garden per year. It doesn't have to take a long time. A lot depends on your climate.