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

Orders on trees and free products of left-ordered groups