  A dynamic coloring of the vertices of a graph $G$ starts with an initial
subset $S$ of colored vertices, with all remaining vertices being non-colored.
At each discrete time interval, a colored vertex with exactly one non-colored
neighbor forces this non-colored neighbor to be colored. The initial set $S$ is
called a forcing set of $G$ if, by iteratively applying the forcing process,
every vertex in $G$ becomes colored. If the initial set $S$ has the added
property that it induces a subgraph of $G$ without isolated vertices, then $S$
is called a total forcing set in $G$. The minimum cardinality of a total
forcing set in $G$ is its total forcing number, denoted $F_t(G)$. We prove that
if $T$ is a tree of order $n \ge 3$ with maximum degree~$\Delta$, then $F_t(T)
\le \frac{1}{\Delta}((\Delta - 1)n + 1)$, and we characterize the infinite
family of trees achieving equality in this bound. We also prove that if $T$ is
a non-trivial tree with $n_1$ leaves, then $F_t(T) \ge n_1$, and we
characterize the infinite family of trees achieving equality in this bound. As
a consequence of this result, the total forcing number of a non-trivial tree is
strictly greater than its forcing number. In particular, we prove that if $T$
is a non-trivial tree, then $F_t(T) \ge F(T)+1$, and we characterize extremal
trees achieving this bound.
