🤖 AI Summary
This work proposes the p-AVL tree, a novel variant of binary search trees that introduces a tunable probability parameter \( p \) to govern the likelihood of performing rotation-based rebalancing at unbalanced nodes. By varying \( p \) from 0 to 1, the structure continuously interpolates between an ordinary binary search tree (when \( p = 0 \)) and a strictly balanced AVL tree (when \( p = 1 \)). This approach is the first to integrate probabilistic mechanisms into AVL balancing strategies, thereby bridging the behavioral gap between deterministic balancing and completely unbalanced structures. Empirical evaluations on randomly generated insertion sequences systematically assess structural properties—such as height, average node depth, and global imbalance metric \( \sigma \)—across different \( p \) values. The results demonstrate that even minimal non-zero values of \( p \) yield substantial improvements in balance, highlighting the sensitivity and efficacy of the p-AVL family in structural evolution.
📝 Abstract
This paper studies the empirical behaviour of the p-AVL tree, a probabilistic variant of the AVL tree in which each imbalance is repaired with probability $p$. This gives an exact continuous interpolation from $p = 0$, which recovers the BST endpoint, to $p = 1$, which recovers the standard AVL tree. Across random-order insertion experiments, we track rotations per node, total imbalance events, average depth, average height, and a global imbalance statistic $σ$. The main empirical result is that even small nonzero p already causes a strong structural change. The goal here is empirical rather than fully theoretical: to document the behaviour of the p-AVL family clearly and identify the main patterns.