A Saw would make faster work of a larger diameter tree, I guess it depends on what type of trees are in your area and whats in your way on the trail.
Axe's are handy for other uses besides chopping downed trees from a road, if you need a large hammer to straighten out steering links and suspension parts on the trail, a single bit axe is a handy tool.
If the tree isn't too big, it's a lot faster to hook up a chain or a winch line and move it... "BH"
I carry an Axe....small at 2 1/2 lbs . In winter or stormy season I throw in a powersaw 24" bar.
I carry a Max Ax. Hated spending the money but it really comes in handy. It's an ax with a pouch of attachments--shovel, rake and pic. Digging fire pits, clearing the ground of debris to set up tent, chopping logs etc.
Also carry a small hack saw. Crushed exhaust sucks!
Another benefit--guys know I am a girl not to be messed with
Female rolling with the 93 YJ SOA Locked and Hard at Play
fold up saw and an ax. if i had to pick one it'd be the ax. primarily because i've got a bum shoulder and after about 30 seconds of sawing it feels like it is on fire. and ax doesnt give me problems. and the ax can be used as a BFH if need be
The ONLY thing I don't like about mine is that it's handle isn't replaceable. HOWEVER, it looks like I'd have to set out with absolute destruction in mind to even ATTEMPT to destroy this handle. I think I paid somewhere around $25 for mine and I feel it's money well spent. It takes an edge easily, stays sharp unless I'm chopping limestone (haha), and has a very well-balanced feel to it overall.
That being said, I also have a small foldable saw as well. They both serve their specific purposes well, and since the saw hardly takes up any room, there's no reason NOT to carry it. I say get them both and cover all your bases. Simple.
I use my saw primarily when setting up my tree stand to clear out any brush that covers my shooting lane, but I keep it in my BOB for emergencies.
There's already a link to the Fiskars camp axe, so here's a link to the foldable saw that I carry: