Folding pruning saws are hand-held saws specifically built for removing small limbs, such as when you cut the limbs off of a staff to smooth it for use as a walking stick. You also can use a pruning saw for some brush clearing and the removal of smaller brush, so long as you use a larger saw instead of the pocket-sized version.
A hand ax is one of the most common trail side cutting tools. It easily fits onto a belt and you can use it to cut through a wide variety of materials, ranging from vegetation to small trees. It is light and easy to handle, which makes it easy to use. You can cut small bushes from the side of the path, or you can chop through small trees which may have fallen across the trail.
Shears range in size from one-handed versions common in home gardening, to those which are double-handled and meant to cut small and medium-sized tree branches and clear brush. You can cut up to a 2 inch thick branch with the larger, two-handed types, which makes them handy for hiking in heavily forested areas no one has cleared for some time.
Clearing knives are long-handled tools with a large blade attached to the end. In some cases the blade is replaced with a saw. You can use them for reaching lower branches of trees that are too high for you to reach by hand. The blade version slices branches, while the saw is for cutting through limbs.
