  Let $G, H$ be two non-empty graphs and $k$ be a positive integer. The
Gallai-Ramsey number $\operatorname{gr}_k(G:H)$ is defined as the minimum
positive integer $N$ such that for all $n\geq N$, every $k$-edge-coloring of
$K_n$ contains either a rainbow subgraph $G$ or a monochromatic subgraph $H$.
The Gallai-Ramsey multiplicity $\operatorname{GM}_k(G:H)$ is defined as the
minimum total number of rainbow subgraphs $G$ and monochromatic subgraphs $H$
for all $k$-edge-colored $K_{\operatorname{gr}_k(G:H)}$. In this paper, we get
some exact values of the Gallai-Ramsey multiplicity for rainbow small trees
versus general monochromatic graphs under a sufficiently large number of
colors. We also study the bipartite Gallai-Ramsey multiplicity.
