+1 on Chris' points about the BCB wire saw. They have done quite well in my tests. You can get them
here from the great folks at Best Glide.
With my GB Small Forest Axe I can easily chop down pretty sizeable trees. Let's say anywhere up to 8" with relatively little effort and a good deal bigger than that if I really had to. Good luck trying to cut down a tree that size with your pocket knife or multitool saw.
Good luck fitting that axe into a palm-sized survival kit.