The Li-Chao tree (LICT) was first introduced in lecture as an efficient data structure for dynamic lower envelope maintenance. In the years since, it has achieved widespread adoption within the competitive programming community, yet no formal specification has appeared in the peer-reviewed literature. This paper provides the definitive formalization of the Li-Chao tree, serving as both the official specification and an expansion of the original lecture. We present complete algorithmic specifications, establish formal correctness proofs, analyze theoretical complexity, and provide empirical performance characterization. The LICT offers distinct advantages in implementation simplicity, numerical stability, and extensibility to advanced variants such as persistence and line segments.

翻译：李超树（LICT）最初在课程中被提出，作为一种用于动态维护下包络线的高效数据结构。自那时起，该结构在算法竞赛社区中得到了广泛采用，但尚未有经过同行评审的正式规范出现在文献中。本文提供了李超树的权威形式化定义，既作为官方规范，也对原始课程内容进行了扩展。我们给出了完整的算法规范，建立了形式化的正确性证明，分析了理论复杂度，并提供了实证性能表征。李超树在实现简洁性、数值稳定性以及对持久化、线段等高级变体的可扩展性方面具有显著优势。