The axe's primary purpose is to cut quite a serious size trees (for fire and shelter), not just splitting some kindling. Sure thing, that's not what a typical overnight walk in the park demands or even allows. But for a survival BoB in rural environment with abundance of trees along the way I would definitely get something like this Fiskars zombie head chopper:
http://au.fiskars.com/Gardening-Yard-Car...hopping-Axe-X15A decent folding saw might be OK for a 2-4 inch twig, but it takes quite a time and effort to saw it, even if it's a dry wood, and the tree is already on the ground. An axe cuts such a piece in 5-10 skillful hits without even breaking a sweat. That's about 10 times faster. Imagine building a full size sleeping wood platform for 2 above the snow with just a folding saw and a knife, it will take half of the day or more. With a long axe - an hour and half - max. Also you don't have to look around for a rare not rotten or dump downed tree trunk for the overnight fire, just find an always crisp dry standing dead tree nearby.
Other points:
- an axe is almost indestructible piece of gear insensitive to abuse, saw and knife are much more fragile tools;
- it's almost impossible to sharpen the saw (at least with the teeth type of folding versions) in the field, an axe can be sharpened with almost any stone.
- an axe is a great hammer for shelter and the camp building tasks.
IMO, an axe is incomparably more effective survival tool than folding saw and a knife all together. But it definitely requires more skill to use in different tasks.
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