Tree Data Structures
Master 4 tree data structures with side-by-side comparisons. Understand hierarchical organization, balance tradeoffs, and real-world applications that power databases, file systems, and compilers.
What Is a Tree? (And Why Engineers Love Them)
A tree is a data structure that organizes data hierarchically. Instead of a flat list, data is arranged in levels:
- A root at the top
- Nodes connected by parent-child relationships
- Branches that grow as data is added
- Leaves at the bottom with no children
Trees are powerful because they reflect real-world hierarchies naturally, reduce search space dramatically, and scale better than linear structures.
Simple intuition: A tree is how you avoid checking everything when you only need one thing.
Where Tree Structures Power Real Systems
Trees quietly run a large part of modern software. Whenever data has levels, trees are usually involved.
Databases & Indexing
- Indexes use balanced trees to find rows quickly
- Without trees, queries degrade to full scans
- B-trees power most database engines
File Systems
- Folders and files form a tree
- Path traversal is tree traversal
- Every directory listing is a subtree
Compilers & Interpreters
- Abstract Syntax Trees (ASTs)
- Code becomes structured data
- Every expression parses into a tree
Search & Autocomplete
- Prefix trees power autocomplete
- Balanced trees enable fast lookup
- No need to scan everything
Why an Unbalanced Tree Is a Performance Trap
Trees promise fast operations only when they stay balanced. A balanced tree keeps its height small. A skewed tree quietly turns into a linked list.
| Tree Shape | Height | Search Time |
|---|---|---|
| Balanced tree | O(log n) | O(log n) |
| Skewed tree (linked list) | O(n) | O(n) |
This is why self-balancing trees exist. AVL Trees rotate aggressively to stay balanced. Splay Trees rebalance based on usage. Heaps enforce structure differently for priority access.
Engineering lesson: Structure without balance is a performance illusion.
Different Trees, Different Tradeoffs
Trees are optimized around different priorities. Some optimize worst-case guarantees. Some optimize average access patterns. Some sacrifice search speed for priority access. There is no universal winner—only tradeoffs.
| Goal | Best Structure |
|---|---|
| Simple implementation | BST |
| Guaranteed fast operations | AVL Tree |
| Fast max or min access | Heap |
| Cache-friendly access patterns | Splay Tree |
How Trees Think About Data
Most tree operations revolve around a few key concepts. Insertions and deletions are not just adds and removes—they are structural changes that preserve invariants.
- Traversal: how you move through nodes
- Comparison: deciding left or right
- Rotation: restoring balance
- Reheapification: maintaining priority
Mental model: Trees constantly reorganize themselves to stay efficient.
Complexity Comparison
| Algorithm | Search | Insert | Delete | Space | Self-Balancing? |
|---|---|---|---|---|---|
| Binary Search Tree | O(h) | O(h) | O(h) | O(n) | |
| AVL Tree | O(log n) | O(log n) | O(log n) | O(n) | |
| Max Heap | O(n) | O(log n) | O(log n) | O(n) | |
| Splay Tree | O(log n) Amortized | O(log n) Amortized | O(log n) Amortized | O(n) |
Click any structure name to read its full article with code examples, history, and use cases.
Quick Decision Guide
Need guaranteed O(log n) operations?
Simple implementation, don't need guarantees?
Frequently accessing recently used elements?
Need a priority queue?
Building a database index?
Implementing Heap Sort?
Common Interview Questions
Why is a balanced tree O(log n)?
Because each level eliminates half of the remaining search space. With n nodes, a balanced tree has log n levels.
When does a BST degrade to O(n)?
When values are inserted in sorted order without rebalancing. The tree becomes a linked list.
This is why AVL trees exist—they rotate to prevent this degradation.
Why are AVL Trees faster for searches but slower for inserts?
Because they perform rotations aggressively to maintain strict balance. Every insert may trigger rebalancing.
Why is heap search O(n)?
Because heaps only guarantee order between parent and children, not across subtrees. You cannot binary search a heap.
Why are splay trees useful?
They move frequently accessed elements closer to the root, optimizing for access patterns. Great for caches and autocomplete.
Myths and Quick Facts
Tree Myths
"BSTs are always fast"
Only if balanced
"Heaps are sorted trees"
They are not—only parent-child order is guaranteed
"AVL trees are always better"
Inserts and deletes are more expensive
"Splay trees are unpredictable"
Amortized performance is still strong
Did You Know?
- Most databases use B-trees or variants, not simple BSTs
- Tree rotations are constant-time operations
- Heaps are usually stored as arrays, not node-pointer structures
- Trees show up everywhere once you start looking
Tree Personalities
BST trusts the input and hopes for the best. AVL is strict and never relaxes. Heap only cares about priority. Splay tree remembers what you like.
See Trees Reshape Themselves
Insert, search, and delete values to watch how trees change shape. Some rebalance aggressively. Some adapt slowly. Some prioritize access over structure. Seeing these changes builds intuition that diagrams cannot.
Open Tree Visualizer