Heights of butterfly trees
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908886579019776 |
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| author | Peca-Medlin, John Zhong, Chenyang |
| author_facet | Peca-Medlin, John Zhong, Chenyang |
| contents | Binary search trees (BSTs) are fundamental data structures whose performance is largely governed by tree height. We introduce a block model for constructing BSTs by embedding internal BSTs into the nodes of an external BST -- a structure motivated by parallel data architectures -- corresponding to composite permutations formed via Kronecker or wreath products. Extending Devroye's result that the height $h_n$ of a random BST satisfies $h_n / \log n \to c^* \approx 4.311$, we show that block BSTs with $nm$ nodes and fixed external size $m$ satisfy $h_{n,m} / \log n \to c^* + h_m$ in distribution. We then study butterfly trees: BSTs with $N = 2^n$ nodes generated from permutations built using iterated Kronecker or wreath products. For simple butterfly trees (from iterated Kronecker products of $S_2$), we give a full distributional description showing polynomial height growth: $\mathbb{E} h_n^{\operatorname{B}} = Θ(N^α)$ with $α= \log_2(3/2) \approx 0.58496$. For nonsimple butterfly trees (from wreath products), we prove power-law bounds: $cN^α\cdot (1 + o(1)) \le \mathbb{E} h_n^{\operatorname{B}} \le dN^β\cdot (1 + o(1))$, with $β\approx 0.913189$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Heights of butterfly trees Peca-Medlin, John Zhong, Chenyang Probability Data Structures and Algorithms Combinatorics Binary search trees (BSTs) are fundamental data structures whose performance is largely governed by tree height. We introduce a block model for constructing BSTs by embedding internal BSTs into the nodes of an external BST -- a structure motivated by parallel data architectures -- corresponding to composite permutations formed via Kronecker or wreath products. Extending Devroye's result that the height $h_n$ of a random BST satisfies $h_n / \log n \to c^* \approx 4.311$, we show that block BSTs with $nm$ nodes and fixed external size $m$ satisfy $h_{n,m} / \log n \to c^* + h_m$ in distribution. We then study butterfly trees: BSTs with $N = 2^n$ nodes generated from permutations built using iterated Kronecker or wreath products. For simple butterfly trees (from iterated Kronecker products of $S_2$), we give a full distributional description showing polynomial height growth: $\mathbb{E} h_n^{\operatorname{B}} = Θ(N^α)$ with $α= \log_2(3/2) \approx 0.58496$. For nonsimple butterfly trees (from wreath products), we prove power-law bounds: $cN^α\cdot (1 + o(1)) \le \mathbb{E} h_n^{\operatorname{B}} \le dN^β\cdot (1 + o(1))$, with $β\approx 0.913189$. |
| title | Heights of butterfly trees |
| topic | Probability Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2507.04505 |