  Given a graph $G$, a total dominating set $D_t$ is a vertex set that every
vertex of $G$ is adjacent to some vertices of $D_t$ and let $d_t(G,i)$ be the
number of all total dominating sets with size $i$. The total domination
polynomial, defined as $D_t(G,x)=\sum\limits_{i=1}^{| V(G)|} d_t(G,i)x^i$,
recently has been one of the considerable extended research in the field of
domination theory. In this paper, we obtain the vertex-reduction and
edge-reduction formulas of total domination polynomials. As consequences, we
give the total domination polynomials for paths and cycles. Additionally, we
determine the sharp upper bounds of total domination polynomials for trees and
characterize the corresponding graphs attaining such bounds. Finally, we use
the reduction-formulas to investigate the relations between vertex sets and
total domination polynomials in $G$.
