I’ve seen and tested my different pocket saws. Most of the saws that are packed in survival kits and Bug Out Bags tend to be the hand chain saw type. These are not the easiest to use (they require a 2-handed approach) but they do take up very little space.
The Folding Saw from Sven-Saw looks like a good option for backpacks. The Sven-Saw has a blade made in Denmark from Swedish steel while the frame is made in the USA.
The saw comes in 2 sizes : 15 inches and 21 inches. It folds down to a pretty compact form factor – most of all, it’s usable with one hand.