A Geometrical Structure for an Infinite Oriented Cluster and its Uniqueness

We consider the supercritical oriented percolation model. Let $\text{fK}$ be all the percolation points. For each $u\in \text{fK}$, we write ${\gamma }_u$ as its right-most path. Let $G={\cup }_u{\gamma }_u$. In this paper, we show that $G$ is a single tree with only one topological end. We also present a relationship between $\text{fK}$ and $G$ and construct a bijection between $\text{fK}$ and $\mathbb{Z}$ using the preorder traversal algorithm. Through applications of this fundamental graph property, we show the uniqueness of an infinite oriented cluster by ignoring finite vertices.