Refined Horton-Strahler numbers I: a discrete bijection
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| Formato: | Preprint |
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2024
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| _version_ | 1866914824778153984 |
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| author | Addario-Berry, Louigi Albenque, Marie Donderwinkel, Serte Khanfir, Robin |
| author_facet | Addario-Berry, Louigi Albenque, Marie Donderwinkel, Serte Khanfir, Robin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03025 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Refined Horton-Strahler numbers I: a discrete bijection Addario-Berry, Louigi Albenque, Marie Donderwinkel, Serte Khanfir, Robin Combinatorics Probability 05C05, 05A19, 05C30 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. |
| title | Refined Horton-Strahler numbers I: a discrete bijection |
| topic | Combinatorics Probability 05C05, 05A19, 05C30 |
| url | https://arxiv.org/abs/2406.03025 |