精细化的Horton-Strahler数 I：一个离散双射
Refined Horton-Strahler numbers I: a discrete bijection
AI总结：
本文提出一种新的双射证明，通过引入插值完全二叉树的树列来定义精细化Horton-Strahler数，证明了具有特定内部顶点数和该数的满二叉树与特定长度和高度的Dyck路径数量相同。
摘要：
有根树$T$的Horton-Strahler数是可以同胚嵌入$T$的最高完全二叉树的高度。已知具有$n$个内部顶点且Horton-Strahler数为$s$的满二叉树的数量，与长度为$2n$且高度$h$满足$\lfloor \log_2(1+h)\rfloor=s$的Dyck路径的数量相同。在本文中，我们给出了上述结果的一个新的双射证明，实际上对该结果进行了加强和精细化，具体如下。我们引入了一列树$(τ_i, i \ge 0)$，它们“插值”了完全二叉树，即对于所有$h \ge 0$，$τ_{2^h-1}$是高度为$h$的完全二叉树，且对于所有$i \ge 0$，$τ_{i+1}$严格包含$τ_i$。定义$\mathcal{S}(T)$为$τ_i$能够同胚嵌入$T$的最大$i$，我们随后证明，具有$n$个内部顶点且$\mathcal{S}(T)=h$的满二叉树$T$的数量，与长度为$2n$且高度为$h$的Dyck路径的数量相同。（我们将$\mathcal{S}(T)$称为$T$的精细化Horton-Strahler数。）我们的证明是双射的，依赖于将二叉树（相应地，Dyck路径）递归分解为具有严格更小的精细化Horton-Strahler数的子树（相应地，具有严格更小高度的子路径）。在后续论文中，我们将证明该双射具有一个连续统类比，它将Brownian连续随机树转化为Brownian excursion，并且在该映射下，树的精细化Horton-Strahler数（的连续类比）成为该excursion的高度。
英文摘要：
The Horton-Strahler number of a rooted tree $T$ is the height of the tallest complete binary tree that can be homeomorphically embedded in $T$. The number of full binary trees with $n$ internal vertices and Horton-Strahler number $s$ is known to be the same as the number of Dyck paths of length $2n$ whose height $h$ satisfies $\lfloor \log_2(1+h)\rfloor=s$. In this paper, we present a new bijective proof of the above result, that in fact strengthens and refines it as follows. We introduce a sequence of trees $(τ_i,i \ge 0)$ which "interpolates" the complete binary trees, in the sense that $τ_{2^h-1}$ is the complete binary tree of height $h$ for all $h \ge 0$, and $τ_{i+1}$ strictly contains $τ_i$ for all $i \ge 0$. Defining $\mathcal{S}(T)$ to be the largest $i$ for which $τ_i$ can be homeomorphically embedded in $T$, we then show that the number of full binary trees $T$ with $n$ internal vertices and with $\mathcal{S}(T)=h$ is the same as the number of Dyck paths of length $2n$ with height $h$. (We call $\mathcal{S}(T)$ the refined Horton-Strahler number of $T$.) Our proof is bijective and relies on a recursive decomposition of binary trees (resp. Dyck paths) into subtrees with strictly smaller refined Horton-Strahler number (resp. subpaths with strictly smaller height). In a subsequent paper, we will show that the bijection has a continuum analogue, which transforms a Brownian continuum random tree into a Brownian excursion and under which (a continuous analogue of) the refined Horton-Strahler number of the tree becomes the height of the excursion.
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