I think I'm right in saying that diamonds are produced by some volcanoes. Why?

They are certainly produced due to volcanic action. Diamonds are formed in diatreme pipes, which are a kind of very steep sided vent down into the Earth. This provides the very high pressure and temperature that diamonds need to form. So it's more of a drilling downwards effect than a coming up and exploding out of a volcano effect. You need the high temperatures, but principally you need the high pressures as well.