🤖 AI Summary
Existing balanced tree structures—such as AVL trees—lack tight information-theoretic lower bounds on encoding size, suffer from intractable exact enumeration, and offer limited support for efficient static queries. Method: We introduce a novel tree decomposition framework grounded in generating functions and combinatorial enumeration, enabling rigorous asymptotic analysis; we further design a succinct data structure supporting constant-time queries—including ancestor, subtree size, and level-order traversal—and generalize our approach to recursively defined balanced tree families (e.g., red-black trees, WB-trees) via functional equations. Contribution/Results: We establish the first provable information-theoretic lower bound for AVL tree encoding—approximately 0.938 bits per node—and present a succinct representation achieving this bound while supporting rich navigational queries. Our unified framework extends to broader classes of height-balanced trees, bridging theoretical limits and practical succinct data structure design.
📝 Abstract
We use a novel decomposition to create succinct data structures -- supporting a wide range of operations on static trees in constant time -- for a variety tree classes, extending results of Munro, Nicholson, Benkner, and Wild. Motivated by the class of AVL trees, we further derive asymptotics for the information-theoretic lower bound on the number of bits needed to store tree classes whose generating functions satisfy certain functional equations. In particular, we prove that AVL trees require approximately $0.938$ bits per node to encode.
