The concept of balance vertices was first investigated by Reid (1999). For the main result “the balance vertices of a tree consist of a single vertex or two adjacent vertices”, Shan and Kang (2004) and Reid and DePalma (2005) improved the length and technique of the proof. In this paper we further discuss the balance vertices on trees in a generalization context. We do not only provide a simple efficient proof for the relevant result but also develop a linear time algorithm to find the set of balance vertices on the underlying tree.

中文翻译：

---

树上平衡顶点的线性时间算法

Reid（1999）首先研究了平衡顶点的概念。对于主要结果“一棵树的平衡顶点由一个顶点或两个相邻顶点组成”，Shan和Kang（2004）以及Reid和DePalma（2005）改进了证明的长度和技巧。在本文中，我们进一步讨论了泛化上下文中树上的平衡顶点。我们不仅为相关结果提供了简单有效的证明，而且开发了线​​性时间算法来找到基础树上的平衡顶点集。