The interleaving distance is a key tool for comparing merge trees, which provide topological summaries of scalar functions. In this work, we define an average merge tree for a pair of merge trees using the interleaving distance. Since such an average is not unique, we propose a method to construct a representative average merge tree. We further prove that the resulting merge tree indeed satisfies a natural notion of averaging for the two given merge trees. To demonstrate the structure of the average merge tree, we include illustrative examples.

翻译：交错距离是用于比较合并树的关键工具，合并树为标量函数提供拓扑摘要。在本研究中，我们利用交错距离定义了一对合并树的平均合并树。由于此类平均树不具唯一性，我们提出了一种构建代表性平均合并树的方法。我们进一步证明所得合并树确实满足对两个给定合并树的自然平均概念。为展示平均合并树的结构，我们提供了说明性示例。