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AVL Trees and the Height-Balanced Alternative

What This Concept Is​

AVL balance requires each node's child heights to differ by at most one. The lab uses empty height 0 and leaf height 1, stores exact height and subtree size, and supports a complete map API including deletion and rank/select. The two metadata fields serve different purposes: height guides rotations; size answers order-statistic queries.

Why It Matters Here​

The invariant is simple enough to check recursively yet strong enough to force logarithmic height. Deletion and metadata repair make this a more demanding exercise than an insertion-only demonstration.

Concrete Examples​

Let N(h) be the minimum number of nodes in an AVL tree of height h. N(0)=0, N(1)=1, and N(h)=1+N(h-1)+N(h-2). A minimally populated height-h node needs one child of height h-1 and another of at least h-2. This Fibonacci-like growth implies h=O(log(n+1)); it is not a measurement of wall-clock lookup speed.

For insertion [3,1,2], root 3 becomes left-heavy but child 1 is right-heavy. Rotate left at 1, then right at 3, obtaining root 2 with children 1,3. Refresh the lower nodes before their new parent. Left-left, right-right and right-left are the other three orientations.

On deletion, a subtree's height can fall and trigger rebalancing repeatedly upward; repairing only the first imbalance is insufficient in general. If the heavy child's two subtree heights tie, a single rotation is appropriate in the usual deletion fixup. Treating equality as the double-rotation case can break balance or do unnecessary work. Test deletion sequences, not only insertion permutations.

For order statistics, size(node)=1+size(left)+size(right). To select index k, compare k with left size; descending right subtracts left size plus one. For rank(x), add that same quantity whenever the search passes a smaller node and proceeds right. Both walk O(log n) nodes when metadata and AVL balance hold.

Common Confusion / Misconceptions​

A tighter theoretical height bound does not establish a universal speed advantage over red-black trees. Allocation, locality, comparison costs and operation mix matter. One single-or-double rebalance after insertion is not a claim that every deletion needs only one. Balanced shape with stale sizes still gives wrong rank/select.

How To Use It​

Validate exact heights, balance, sizes, global key bounds and unique node identity after every mutation. Compare public behavior to sorted dictionary items. Include all four rotations, replacements, root deletion and a two-child deletion with a successor payload. Analyze lookup/update path length separately from O(n) traversal output.

Check Yourself​

Why must size be recomputed even if height stays unchanged? How does the minimum-node recurrence prove logarithmic height? Why can deletion repair propagate?

Mini Drill or Application​

For root 2 with leaves 1 and 3, delete 1 and then replace key 3. Feedback: two nodes remain after deletion; replacement preserves that count. Select index 1 must return key 3 and its new value.

Read This Only If Stuck​

CLRS and Skiena: balanced-tree invariants and rotations. The lab specifies AVL conventions directly, so no particular edition must supply identical code.