I've got an industrial backhoe (may not matter). I've pulled a down tree before and wrapped a chain around the rear axle. I've also pulled a downed tree wrapping the chain around the front bucket, letting the chain run UNDER the machine so that everything is working in the 'pull' position.
Depending on how big/heavy... I've also clamped backhoe around a tree and carried it, much like a dog might carry a large stick "sideways" in its mouth....but to get through some tight area, the dog might turn his head (rotate backhoe) to bring tree more parallel to direction of travel.
I THINK I've also wrapped chain around front bucket to pull.... when chain came out from behind, I tried to hook it with backhoe so I could lift the chain up a bit.
Many years ago, I was pulling a downed tree with grade 100 chain wrapped around axle. When I got to destination, I saw that I literally scraped about have the 3/8th's link in half. (turns out it was caught under weight of tree and was being ground into pavement)
Above might be stupid & dangerous (isn't everything?) I always went slow and (knocking on wood) so far, nothing negative happened.