You say that wood decomposes under heating and produces gases(1) which then burn. That's correct. It would be wrong, though, to assume that all combustible materials are like that. Indeed, many - arguably, most of them are. Wax, tar, and all kinds of plastic come to mind. But carbon is a different story.
See, carbon is very heat-resistant, about as much so as the most heat-resistant thing you know, only more. You heat it to $1000\;^\circ\rm C$, and nothing happens. Well, it may start burning provided there is air around, but short of that, it is nothing. You heat it a few hundred degrees more, it starts glowing red, then white, then steel melts, then you pass $2000\;^\circ\rm C$ and continue to push on, then everything else melts, by now your carbon glows so bright that you can't see a thing, but it is still intact.
Burning of carbon, unlike burning of many other materials, is a true heterophase reaction. You chip atoms off that piece one by one, hence no smoke is produced. Burning of wood or plastic is another story. Combustible gases are produced and fly away, and when some of them do not burn completely(2), they leave a solid residue (which, BTW, may or may not consist mostly of carbon). That's what we call smoke.
P.S. Thought I'd rather add two clarification notes:
(1) When heated, plastic turns into gases all but completely. Wood doesn't. Much of it remains in the form of charcoal, which is mostly carbon and burns like carbon, that is, producing no smoke.
(2) Incomplete combustion of carbon is also possible, and it results in carbon monoxide, which killed quite a number of people over the past few millenia. But it isn't smoke; it's a gas. You can't see it. You just see people dropping dead.
So it goes.