Merge trees are a common topological descriptor for data with a hierarchical component, such as terrains and scalar fields. The interleaving distance, in turn, is a common distance for comparing merge trees. However, the interleaving distance for merge trees is solely based on the hierarchical structure, and disregards any other geometrical or topological properties that might be present in the underlying data. Furthermore, the interleaving distance is NP-hard to compute. In this paper, we introduce a form of ordered merge trees that can capture intrinsic order present in the data. We further define a natural variant of the interleaving distance, the monotone interleaving distance, which is an order-preserving distance for ordered merge trees. Analogously to the regular interleaving distance for merge trees, we show that the monotone variant has three equivalent definitions in terms of two maps, a single map, or a labelling. Furthermore, we establish a connection between the monotone interleaving distance of ordered merge trees and the Fr\'echet distance of 1D curves. As a result, the monotone interleaving distance between two ordered merge trees can be computed exactly in near-quadratic time in their complexity. The connection between the monotone interleaving distance and the Fr\'echet distance builds a new bridge between the fields of topological data analysis, where interleaving distances are a common tool, and computational geometry, where Fr\'echet distances are studied extensively.

翻译：合并树是描述具有层次结构数据（如地形和标量场）的常用拓扑描述符。交错距离则是比较合并树的常用度量。然而，合并树的交错距离仅基于层次结构，忽略了底层数据中可能存在的其他几何或拓扑性质。此外，交错距离的计算是NP难问题。本文提出一种能够捕捉数据内在顺序的有序合并树形式，并进一步定义了交错距离的自然变体——单调交错距离，作为有序合并树的保序度量。类比于常规合并树交错距离，我们证明该单调变体具有三种等价定义，可分别通过双映射、单映射或标注方式表述。更重要的是，我们建立了有序合并树的单调交错距离与一维曲线弗雷歇距离之间的理论关联。基于此，两个有序合并树之间的单调交错距离可在接近平方时间复杂度内精确计算，其复杂度取决于树结构规模。单调交错距离与弗雷歇距离的关联，在拓扑数据分析（交错距离是常用工具）与计算几何（弗雷歇距离被广泛研究）两大领域之间构建了新的桥梁。