The randomly colored graph $G_c(n,p)$ is derived from the Erd\H{o}sR\'{e}nyi binomial random graph $G(n,p)$ by assigning a set color to each edge obtained by assigning Randomly randomize the color of $c$ independently. It is easy to see that when $c = \Theta(n)$, the order of the maximum rainbow tree in this model undergoes a phase transition at the critical point $p=\frac{1}{n}$. In this paper, for some $\varepsilon=\varepsilon, when $p = \frac{1+\varepsilon}{n}$, the number of maximum rainbow trees in \emph{weak subcritical and supercritical regions} Determine the asymptotic order. (n)$ satisfying $\varepsilon = o(1)$ and $|\varepsilon|^3 n\to\infty$. In particular, we show that for both of these regimes, the largest component of $G_c(n,p)$ contains nearly a spanning rainbow tree with high probability. When $p = \frac{d}{n}$ and constant $d >1$, we also consider the order of the largest rainbow tree in the \emph{sparse region}. Here, the largest rainbow tree has a linear order, and furthermore, given that $d$ and $c$ are large enough, there is a high probability that the rainbow cycle nearly spans $G_c(n,p)$ indicates that it contains