Let S be a subset of V(G). The vertices of S are colored black and the vertices of \(V(G)-S\) are colored white. The color-change-rule is defined as if a black vertex u has a unique white neighbor v, then we change the color of v from white to black. The initial S is called a zero forcing set if all vertices of V become black by iteratively applying the color-change-rule above. We call S a connected forcing set (resp. total forcing set) of G if G[S] is a connected subgraph (resp. a subgraph without isolated vertices). The minimum cardinality of zero forcing sets, connected forcing sets and total forcing sets are called zero forcing number, connected forcing number and total forcing number, denoted by F(G), \(F_c(G)\) and \(F_t(G)\) , respectively. Observe that \( F(G)\le F_t(G)\le F_c(G)\) for a connected graph G, in particular, \( F(T)+1\le F_t(T)\le F_c(T)\) for a tree T. In the paper, for a tree T we obtain a sharp upper bound of \(F_c(T)-F_t(T)\) . In addition, we also characterize the structure of T with \(F_c(T)=F_t(T)\) and \(F(T)+1=F_t(T)=F_c(T)\) , respectively.
