For years I have carried a great trail saw on my saddle that I got from Sawtooth Outfitter. They no longer carry them but I did find out where you can get them. The one I like is the Fanno K24 curved tree saw. You can make a saw sheath or the are available to order. Here a a good video on care of your saw.

Fanno Saw Works
Address: 224 W 8th Ave, Chico, CA 95926
Phone:(530) 895-1762
http://www.fannosaw.com/