Balanced Search Trees

Binary search trees give O(log n) lookup, insert, delete — when balanced. Without rebalancing, a BST can degenerate to a linked list (O(n)).

Balanced search trees maintain logarithmic height through rebalancing operations. They're the workhorses of databases, language libraries, and ordered-data structures.

Why balance matters

A BST built by inserting sorted data degenerates:

1
 \
  2
   \
    3
     \
      4

All operations become O(n). Rebalancing keeps height O(log n).

AVL trees

The first balanced BST (1962). Strict balance: heights of subtrees differ by at most 1.

Operations

• Insert: standard BST insert; rebalance up the path
• Delete: standard BST delete; rebalance up the path
• Lookup: O(log n)

Rebalancing uses rotations: single (LL, RR) or double (LR, RL).

Properties

• Height: ≤ 1.44 × log₂(n)
• Stricter balance than red-black
• More rotations on insert/delete
• Faster lookups than red-black (slightly shorter)

Use cases

• Read-heavy workloads (frequent lookups)
• Memory databases

Red-black trees

More relaxed balance. Each node is red or black; properties enforce approximate balance.

Properties

• Root is black
• Red nodes can't have red children
• All paths from root to null have same number of black nodes

These rules guarantee height ≤ 2 × log₂(n+1).

Operations

Same as AVL: insert, delete, lookup. Rebalancing uses rotations and color flips.

Generally fewer rotations than AVL on writes.

Use cases

• Java's TreeMap, TreeSet
• C++'s std::map, std::set
• Linux kernel (process scheduling, mmap)

The default ordered-map implementation in many standard libraries.

B-trees

Multi-way trees. Each node holds k keys and has k+1 children.

Designed for disk-based storage; adapted for in-memory use too.

Properties

• All leaves at same depth
• Each internal node has between m/2 and m children
• m typically 100-1000 for disk-based; smaller for memory

Operations

• Lookup: traverse, comparing keys at each node
• Insert: insert at leaf; split if too full; propagate up
• Delete: more complex; merge or borrow from siblings

B+ trees

All data in leaves; internal nodes have only keys. Leaves linked for efficient range scans.

Standard for relational databases (MySQL, PostgreSQL indices).

Use cases

• Database indices
• Filesystems (ext4, NTFS, btrfs)
• Key-value stores (LevelDB-derived)

Splay trees

Self-adjusting tree. Recently accessed elements move to root.

Properties

• No explicit balance
• Amortized O(log n) operations
• Specific access patterns can be O(log n) per operation

Use cases

Niche. Useful when access patterns are skewed (some keys much hotter).

Less common in production than AVL or red-black.

Treaps

Randomized BST. Each node has a key and a random priority. Tree is BST on keys, heap on priorities.

Properties

• Expected O(log n) operations
• Simple implementation
• Good for concurrent operations (less coordination)

Skip lists

Probabilistic alternative. Multiple levels of linked lists; higher levels are "express lanes."

Properties

• Expected O(log n) operations
• Simpler than balanced BSTs
• Good for concurrent access

Use cases

• Redis (sorted sets)
• LevelDB memtable
• Some concurrent data structures

Choice in practice

For most language standard libraries: red-black trees.

For database indices on disk: B+ trees.

For sorted sets with frequent updates: skip lists or red-black.

Rarely should you implement these yourself. Use the standard library.

Comparison

	AVL	Red-Black	B-tree	Skip List
Lookup	Fast	Slightly slower	Slow per node, few nodes	Fast
Insert	Slower	Fast	Fast	Fast
Memory	Compact	Compact	Compact	More overhead
Cache friendly	Poor	Poor	Excellent	Poor
Concurrent	Hard	Hard	Hard	Easier

Key insight: cache locality

Modern CPUs make cache miss expensive. B-trees with high fan-out are cache-friendly: more comparisons per cache line.

Binary trees have poor cache behavior. This is why B-trees beat binary trees for in-memory ordered data too, despite higher complexity.

For pure performance on modern hardware, B-tree variants often beat AVL/red-black even in memory.

Operations performance

For all balanced BSTs:

• Lookup: O(log n)
• Insert: O(log n)
• Delete: O(log n)
• Min/max: O(log n)
• Range query: O(log n + k) where k is range size
• In-order traversal: O(n)

These are usually 2-3x slower than hash tables for individual lookups, but support ordered operations hash tables can't.

When to use a balanced BST vs hash table

Hash table wins

• Order doesn't matter
• Maximum lookup speed needed
• No range queries

Balanced BST wins

• Order matters
• Range queries
• Predecessor/successor queries
• Worst-case bounds matter (hash tables have O(n) worst case)
• Iterator stability across modifications

Common operations

Range queries

"Find all values in [a, b]"

In balanced BST: O(log n + k). Easy.

In hash table: O(n). Painful.

Predecessor / successor

"Largest value less than x"

In balanced BST: O(log n).

In hash table: O(n).

Order statistics

"What's the k-th smallest value?"

With augmentation (size info per node): O(log n).

Common failure patterns

Implementing your own

Bugs are subtle. Use the standard library.

Hash table when ordering matters

Realize too late that you need ordered iteration.

Balanced BST when hash table sufficient

Slower than hash table for simple lookups.

Ignoring cache effects

Choosing for asymptotic complexity when constant factors dominate at your scale.

Using the wrong balanced BST variant

Almost always the standard library default is fine.

Practical advice

For application code:

• Use the standard library's ordered map (TreeMap, std::map, BTreeMap in Rust)
• Don't worry about which balanced tree it uses
• Benchmark if performance matters

For systems / database code:

• B-tree variants are usually right
• Concurrency considerations matter
• Profile actual workloads

For learning:

• Implement red-black or AVL once for understanding
• Then use the standard library

Further Reading

• HashTableDesign — Alternative for unordered data
• HeapAndPriorityQueues — Different tree structure
• Interval Trees — Augmenting a balanced BST with a max field for O(log n) overlap queries
• Data Structures Hub — Cluster index