I recently bought and recommend this Tarvol 14" curved blade hand saw that comes with a sheeth and is low cost. I can carry it in a small USWE pack and I can use the saw for cutting trees up that have fallen across mtb trails and for doing other minor trail work. A smaller saw that can fit in your pocket or a hip pack, that I also purchased is the Silky Professional Series PocketBoy Folding Saw 130mm Large Teeth and I'm super impressed by how well the Pocketboy saw cuts. It's more expensive, but a really well made saw.
If you're going to mountain bike with a saw, I think it's a must to get one with a sheeth to keep the saw blade cutting you (or a folding saw), or it ripping a camelbak bladder or backpack.
The Tarvol hand saw cuts really well and is light weight. I've probably cut downed trees that were up to 12" in diameter. You could probably cut even larger trees, but it might take you a while with a hand saw. In that case, you might think about getting a small chainsaw.
I literally don't even notice that the saw is in my pack while riding, and it sticks part way out the zipper of the pack. So all in all, for the price I'd definitely pick one up... and as a bonus you can use it to prune branches around your house or cut down a fresh Christmas tree during the holidays!