In this work, we study the color discrepancy of spanning trees in random graphs. We show that for the Erdős-Rényi random graph $G(n,p)$ with $p$ above the connectivity threshold, the following holds with high probability: in every 2-edge-coloring of the graph, there exists a spanning tree with a linear number of leaves such that one color class contains more than $\frac{1 + \varepsilon}{2}n $ of the tree's edges. Here, $\varepsilon>0$ is a small absolute constant independent of $p$. We also extend this line of research to randomly perturbed dense graphs, showing that adding a few random edges to a dense graph typically creates a spanning tree with a large color discrepancy under any 2-edge-coloring.