  In this paper, we study the differentiation operator acting on discrete
function spaces; that is spaces of functions defined on an infinite rooted
tree. We discuss, through its connection with composition operators, the
boundedness and compactness of this operator. In addition, we discuss the
operator norm and spectrum, and consider when such an operator can be an
isometry. We then apply these results to the operator acting on the discrete
Lipschitz space and weighted Banach spaces, as well as the Hardy spaces defined
on homogeneous trees.
