For the wood shader, do we have to have specular reflections?

It's your choice. Some types of wood have specular reflections, some types of wood do not. Note that, if you call your diffuse plus specular shader (i.e.

Your red plastic shader) from inside your wood shader, you can get both diffuse and specular reflections without writing any new code. Just have your wood shader set the diffuse color (and possibly also the specular color) before calling your diffuse plus specular shader to finish off the lighting computation. More.

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