  Cameron, et al. determined the maximum size of a null subsemigroup of the
full transformation semigroup $\mathcal{T}(X)$ on a finite set $X$ and provided
a description of the null semigroups that achieve that size. In this paper we
extend the results on null semigroups (which are commutative) to commutative
nilpotent semigroups. Using a mixture of algebraic and combinatorial
techniques, we show that, when $X$ is finite, the maximum order of a
commutative nilpotent subsemigroup of $\mathcal{T}(X)$ is equal to the maximum
order of a null subsemigroup of $\mathcal{T}(X)$ and we prove that the largest
commutative nilpotent subsemigroups of $\mathcal{T}(X)$ are the null semigroups
previoulsy characterized by Cameron, et al..
