Woods & Stuff
Usual disclaimer: "woods" means different things to different people. The woods in Alaska, Idaho, the northern Sierra Nevadas, and the woods found in the Rockies are all gonna be different, and also different from, say, the Appalachian woods.
So, with that out of the way, let me try to put down something reasonably coherent.
In most situations, I'm a knife-and-hatchet guy, but there are any number of exceptions.
The woods around here in North Idaho? I would normally take a decent fixed blade, a good folder, and a hatchet. The hatchet would be a light one, either something worn on the belt or affixed to the pack. A Gerber/Fiskars hatchet, or maybe the ultra-light Trail Blazer hatchet (made in Canada). The larger Trail Blazer with the bent handle offers a good combination of cutting edge, lightness, and application ergos. The blade is flat ground, the way a knife would be, which offers some flexibility and ease of sharpening.
Also up in this area there are places with thicker brush, and I might switch over to a short-bladed survival machete, like the Ontario SP8. It will do hatchet duty, and it's better suited to brush clearing than a hatchet.
If I were farther north, and game was involved, I might prefer something like the Timberline/Kommer Bush Pilot hatchet. It's designed to allow you to dress and skin game, quite in addition to mundane camp duties.
If I were in California/Nevada up in the Sierras, I would go for a Woodman's Pal, a light machete, or a Khukri. The Sierras have Manzanita, and that stuff requires a blade with a good cutting edge and a bit of a hook. Hatchets need not apply.
Back East, I would probably select something similar, but I haven't been in those woods in several decades.
High country in Arizona/New Mexico? Probably a machete or Khukri. My dad used to bring his Filipino bolo knife. Lightweight, medium length, awesome cutting edge. It would cut through even the springy stuff, cactus, the thorny trees whose name I've just now forgotten, and stuff like that. I would imagine a well balanced Khukri would perform equally well.
What I would *not* bring to the woods -- any woods of my acquaintance -- is a random full-length machete made for working in long grass or cane. Steel is too soft for hacking around in the woods, and the extra length would just get in the way.
Now that I think a little more about it, I would be completely comfortable substituting my dad's Filipino bolo for a hatchet in most circumstances. That thing even worked on Manzanita.
Okay, now I actually need to do some research and see if I can find one of those. How did I forget that?
So, yeah, a well balanced Khukri, Filipino style bolo, . . . something like that . . . would do okay in the woods.
That said, you should remember that if you're going to do any serious amount of chopping, that a hatchet or camp axe is actually balanced and weighted for chopping. I mean, think, there's a reason lumberjacks use axes for choppage. The tool is optimized for that.
The more your requirement involves versatility, the more it favors a light, robust, bolo/Khukri/short machete.