Heights of butterfly trees
John Peca-Medlin, Chenyang Zhong

TL;DR
This paper introduces a block model for binary search trees (BSTs) inspired by parallel data architectures, analyzes their heights, and explores butterfly trees generated by Kronecker and wreath products, revealing polynomial and power-law growth behaviors.
Contribution
It extends existing BST height results to block models and provides a detailed distributional analysis of butterfly trees with different generative methods.
Findings
Block BSTs have height ratios converging to a sum of constants.
Simple butterfly trees exhibit polynomial height growth with exponent ~0.585.
Nonsimple butterfly trees have height bounds following power-law behavior.
Abstract
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 of a random BST satisfies , we show that block BSTs with nodes and fixed external size satisfy in distribution. We then study butterfly trees: BSTs with nodes generated from permutations built using iterated Kronecker or wreath products. For simple butterfly trees (from iterated Kronecker products of ), we give a full distributional description showing polynomial height growth: $\mathbb{E}…
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Taxonomy
TopicsAfrican Botany and Ecology Studies
