  Given graphs $H_1, \dots, H_t$, a graph $G$ is $(H_1, \dots,
H_t)$-Ramsey-minimal if every $t$-coloring of the edges of $G$ contains a
monochromatic $H_i$ in color $i$ for some $i\in\{1, \dots, t\}$, but any proper
subgraph of $G $ does not possess this property. We define
$\mathcal{R}_{\min}(H_1, \dots, H_t)$ to be the family of $(H_1, \dots,
H_t)$-Ramsey-minimal graphs. A graph $G$ is \dfn{$\mathcal{R}_{\min}(H_1,
\dots, H_t)$-saturated} if no element of $\mathcal{R}_{\min}(H_1, \dots, H_t)$
is a subgraph of $G$, but for any edge $e$ in $\overline{G}$, some element of
$\mathcal{R}_{\min}(H_1, \dots, H_t)$ is a subgraph of $G + e$. We define
$sat(n, \mathcal{R}_{\min}(H_1, \dots, H_t))$ to be the minimum number of edges
over all $\mathcal{R}_{\min}(H_1, \dots, H_t)$-saturated graphs on $n$
vertices. In 1987, Hanson and Toft conjectured that $sat(n,
\mathcal{R}_{\min}(K_{k_1}, \dots, K_{k_t}) )= (r - 2)(n - r + 2)+\binom{r -
2}{2} $ for $n \ge r$, where $r=r(K_{k_1}, \dots, K_{k_t})$ is the classical
Ramsey number for complete graphs. The first non-trivial case of Hanson and
Toft's conjecture for sufficiently large $n$ was setteled in 2011, and is so
far the only settled case. Motivated by Hanson and Toft's conjecture, we study
the minimum number of edges over all $\mathcal{R}_{\min}(K_3,
\mathcal{T}_k)$-saturated graphs on $n$ vertices, where $\mathcal{T}_k$ is the
family of all trees on $k$ vertices. We show that for $n \ge 18$, $sat(n,
\mathcal{R}_{\min}(K_3, \mathcal{T}_4)) =\lfloor {5n}/{2}\rfloor$. For $k \ge
5$ and $n \ge 2k + (\lceil k/2 \rceil +1) \lceil k/2 \rceil -2$, we obtain an
asymptotic bound for $sat(n, \mathcal{R}_{\min}(K_3, \mathcal{T}_k))$.
