I mean, I'm sure we all know the most common chainsaw injuries. Random numbers I found (cited as way back in 1994...but I'm sure the current trends are the same)
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Obviously from (by the numbers) one handing, dropping the saw into your leg while standing, kickback to the head, etc.
This is probably everyone, not just pros, so I bet there aren't many climbing incidents there. Certainly because the body position is so different, common injuries will be as well. My guess would be left hand, then kickback into the face/neck.
Chainsaw pants just mean you are protecting one of the most hit areas, and it's the easiest part to add protection to. By simply wearing them, you just know you are always protected (obviously nothing is 100%) in that area at all times, no need to run back to the truck and grab some chaps to make one or two cuts, which keeps you from skipping that step.