二叉树高度的热带代数
The Tropical Algebra of Binary-Tree Height
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中文总结 AI 辅助
本文研究二叉树高度的热带代数$\boldsymbol{\textit{H}}$，证明带标记树加权求值仅与标记最大深度有关，明确其单树轮廓、自由代数结构，分类相关代数结构并恢复二进合成谱。
中文摘要 AI 辅助
二叉树高度递归在自然数集与负无穷的并集上定义了一个代数$\boldsymbol{\textit{H}}$，其交运算为$\boldsymbol{\textit{max}}$，乘积为$\boldsymbol{a\bigstar b = max\{a,b\} + 1}$。我们证明带标记树的加权求值仅取决于每个标记的最大深度。单树轮廓恰好是满足二叉Kraft不等式的向量，而有限交可实现$\boldsymbol{(\boldsymbol{\textit{N}}\boldsymbol{\bigcup}\boldsymbol{\big{-}}\boldsymbol{\textit{\textbackslash infty}\big{)}}^n}$中的每个向量；因此$n$变量项运算构成自由代数$\boldsymbol{\textit{H}}^n$。我们还分类了$\boldsymbol{\textit{H}}$的兼容半格运算、子代数、自同态、同余和有限商，并在全线性边界上恢复了二进合成谱。
英文摘要
The binary-tree height recursion defines an algebra $\mathcal{H}$ on $\mathbb{N}\cup\{-\infty\}$, with join given by $\max$ and product \[ a\star b=\max\{a,b\}+1. \] We show that weighted evaluation of a labelled tree depends only on the greatest depth of each label. Single-tree profiles are exactly the vectors satisfying the binary Kraft inequality, while finite joins realize every vector in $\bigl(\mathbb{N}\cup\{-\infty\}\bigr)^n$; hence the $n$-variable term operations form the free algebra $\mathcal{H}^n$. We also classify $\mathcal{H}$'s compatible semilattice operation, subalgebras, endomorphisms, congruences, and finite quotients, and recover the dyadic-composition spectrum at the full-linear boundary.
