yes they do... but it takes a long time. A girdled tree can live for one or two seasons before it completely dies (depending upon the weather and the size of the tree).

Once it dies, it then becomes home for insects and other critters and the woodpeckers will start to go at it also. Eventually they rot and fall over, but it does take some time.

If you want the tree for fuel or firewood, it would probably be okay standing for one or two seasons, but I tend to think that after that you really should cut, chop, split and stack to get the most out of it for firewood.

Just take the trees down, don't drag it out. Buck, cut, split, whatever, then tarp it.

Nature attacks anything that is weak. If you leave that dying timber standing, it will be worth less every day.