I was thinking that Doug Ritter once said that a large handled ax is a required part of a Canadian or Alaska pilot's supplies. I may not have that right though. Per the comments, I always wondered what that axe was for.

To me, axes are heavy and relatively dangerous under an assumption that there will be limited emergency healthcare. Even in a non survival world, I really only use axes for splitting wood. A saw works much faster and easier for crosscutting.

In an emergency situation, for fire it it is easier and safer to use small wood. It might be useful to have a small saw to cut standing (dry) wood, but small saws like the folding saws tend to be all too "bendable". My experience has been that a bent saw can become almost useless.

Bow-type saws are much more durable, but much bigger. Years back many used a Sven folding saw, but I found its sharp corners to be worrisome. We used to wrap the edges with cloth - that worked At one time I like something called a Takedown Bucksaw - or similar, but I'm not sure that is still around. My problem with that is that it used a very easy to lose clip - a bad design in my mind. There appear to be many folding bow saws on the market now, but I'm not familiar with them.

There are also the wire-type saws, but in general I haven't heard good things about them. I've used the pocket chainsaw, and it actually cut very well, but I was always concerned about its complexity. Too many joints to fail.

For shelter, a saw could certainly help a lot.