We construct total orders on the vertex set of an oriented tree. The orders are based only on up-down counts at the interior vertices and the edges along the unique geodesic from a given vertex to another. As an application, we provide a short proof (modulo Bass-Serre theory) of Vinogradov's result that the free product of left-orderable groups is left-orderable.

中文翻译：

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左序群体的树木和免费产品的订单

我们在有向树的顶点集上构造总阶数。顺序仅基于内部顶点和沿唯一测地线从给定顶点到另一个顶点的上下计数。作为应用，我们提供了 Vinogradov 结果的简短证明（模 Bass-Serre 理论），即左序群的自由积是左序的。