  Given an integer $r\ge1$ and graphs $G, H_1, \ldots, H_r$, we write $G
\rightarrow ({H}_1, \ldots, {H}_r)$ if every $r$-coloring of the edges of $G$
contains a monochromatic copy of $H_i$ in color $i$ for some $i\in\{1, \ldots,
r\}$. A non-complete graph $G$ is $(H_1, \ldots, H_r)$-co-critical if $G
\nrightarrow ({H}_1, \ldots, {H}_r)$, but $G+e\rightarrow ({H}_1, \ldots,
{H}_r)$ for every edge $e$ in $\overline{G}$. In this paper, motivated by
Hanson and Toft's conjecture [Edge-colored saturated graphs, J Graph Theory
11(1987), 191--196], we study the minimum number of edges over all $(K_t,
\mathcal{T}_k)$-co-critical graphs on $n$ vertices, where $\mathcal{T}_k$
denotes the family of all trees on $k$ vertices. Following Day [Saturated
graphs of prescribed minimum degree, Combin. Probab. Comput. 26 (2017),
201--207], we apply graph bootstrap percolation on a not necessarily
$K_t$-saturated graph to prove that for all $t\ge4 $ and $k\ge \max\{6, t\}$,
there exists a constant $c(t, k)$ such that, for all $n \ge (t-1)(k-1)+1$, if
$G$ is a $(K_t, \mathcal{T}_k)$-co-critical graph on $n$ vertices, then $$
e(G)\ge \left(\frac{4t-9}{2}+\frac{1}{2}\left\lceil \frac{k}{2}
\right\rceil\right)n-c(t, k).$$ Furthermore, this linear bound is
asymptotically best possible when $t\in\{4,5\}$ and $k\ge6$. The method we
develop in this paper may shed some light on attacking Hanson and Toft's
conjecture.
