Random sequential nearest-neighbor coloring on trees

We study a random sequential coloring process on finite regular trees, where the root is initially colored in blue and the leaves in red. The remaining vertices are selected uniformly at random and colored sequentially, each vertex inheriting the color of its closest previously colored vertex. We prove that, as the height of the tree tends to infinity, blue vertices appear arbitrarily close to the leaves with high probability, while red vertices persist at bounded distance from the root. We also show that this procedure yields a non-degenerate infinite-volume limit on the infinite regular tree, in contrast with the Euclidean Poissonian coloring.