Heights of butterfly trees

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 $h_n$ 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: $\mathbb{E} h_n^{\operatorname{B}} = \Theta(N^\alpha)$ with $\alpha = \log_2(3/2) \approx 0.58496$. For nonsimple butterfly trees (from wreath products), we prove power-law bounds: $cN^\alpha\cdot (1 + o(1)) \le \mathbb{E} h_n^{\operatorname{B}} \le dN^\beta\cdot (1 + o(1))$, with $\beta \approx 0.913189$.