Hi Harry,
Well, it really depends on the size of the branch which determines what you should use (lets face it, the least amount of work is surely the best?!)
Also, you should take into account the damage that can be done to the tree in the first place. If you use the back and forth action of a saw, it leaves a lot of serrated edges on the wood leaving it open for infections while an axe leaves a very smooth surface which seals a lot more quickly.
Another tool that can be used and is often as effective as an axe is the billhook..a much less bulky and portable tool.
Hope that helps.
Jamie