Paper is used to provide dryness and absorb water from the manure based and other composts. Newspaper is the best source for this material as it easily absorbs the moisture from the manure mixture providing some consistency in the product.
You can't really use hard wood or a Christmas tree without first pulping the tree and reducing it to fiber. The wood fiber from a tree will not have the same absorbent qualities as paper. Turning trees into mulch is much more efficient than pulping. Pulping takes a fair amount of energy.