Heights of butterfly trees
This work provides theoretical insights into tree height for specialized BST structures, which is incremental for data structure theory and parallel computing applications.
The paper tackles the problem of analyzing the height growth of binary search trees (BSTs) constructed using block models and butterfly trees, showing that simple butterfly trees exhibit polynomial height growth with exponent α ≈ 0.58496, and nonsimple ones have power-law bounds with exponents up to β ≈ 0.913189.
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: $\mathbb{E} h_n^{\operatorname{B}} = Î(N^α)$ 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 $β\approx 0.913189$.