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DSA SERIES Chapter 10 -INTRODUCTION TO TREES
1. What Is a Tree?
A Tree is a non-linear, hierarchical data structure consisting of nodes connected by edges. Unlike arrays, stacks, queues, or linked lists, a tree does not store data sequentially.
Instead, it organizes data in parent–child relationships.
A tree with N nodes always contains N – 1 edges, ensuring a single connected structure without cycles.
2. Why Trees?
Trees serve as the backbone for numerous critical algorithms and advanced data structures.
Reasons why trees are indispensable in computer science:
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Represent hierarchical relationships
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Faster searching compared to linked lists
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Support sorted operations (BST)
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Basis for self-balancing structures (AVL, Red-Black Trees)
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Enable memory-efficient representation of sparse data
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Form the foundation of:
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Heaps
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Tries
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Segment Trees
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Huffman Trees
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Expression Trees (Compilers & Interpreters)
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Trees help bridge theoretical data structures with practical system usage.
3. Key Terminology
Node
Basic unit containing data, and links to its children.
Root
Topmost node with no parent.
Every tree has exactly one root.
Parent
A node that has one or more children.
Child
Nodes directly connected below a parent.
Siblings
Children of the same parent.
Leaf / External Node
Node with zero children.
Internal Node
Node with at least one child.
Ancestor
Any node on the path from the root to a given node.
Descendant
Any node reachable downward from a given node.
Degree of a Node
Number of children a node has.
Height of a Node
Longest downward path from the node to a leaf.
Height of a Tree
Height of the root (maximum depth of any node).
Depth of a Node
Distance (in edges) from the root to that node.
Subtree
A tree formed by any node and its descendants.
4. Properties of Trees
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Tree with N nodes has N-1 edges.
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Root has no parent.
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Leaves have no children.
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Exactly one unique path exists between any two nodes.
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Trees do not contain cycles.
5. Types of Trees
Important categories:
General Tree
Nodes can have any number of children.
Binary Tree
Each node can have 0, 1, or 2 children.
Binary Search Tree (BST)
Left child < parent < right child.
Balanced Trees
Maintain height ≈ log n
Examples: AVL, Red-Black Trees.
Heaps
Binary tree used for priority queues.
Expression Trees
Used in compilers for evaluating expressions.
Tries
Prefix tree used for dictionaries, autocomplete, and spell-checking.
Segment Trees / Fenwick Trees
Used in competitive programming for fast range queries.
6. Real-World Applications
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File system hierarchy
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Database indexing (B-Trees, B+ Trees)
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Network routing algorithms
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Evolutionary trees in bioinformatics
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AI search algorithms (Min-Max Trees)
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Syntax trees in compilers
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Operating system process trees
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Game development states and scene graphs
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Scheduling and resource allocation
7. Differences: Tree vs Graph
| Feature | Tree | Graph |
|---|---|---|
| Cycles | No | May contain |
| Root | One | Not necessary |
| Edge Count | N – 1 | Arbitrary |
| Unique Path | Yes | Not guaranteed |
| Traversals | DFS, BFS | DFS, BFS + many others |
Trees are a special case of graphs (acyclic, connected, directed/undirected rooted structure).
8. Operations on Trees
Standard operations include:
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Insertion
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Deletion
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Searching
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Traversal
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Inorder
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Preorder
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Postorder
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Level order
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Height computation
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Counting leaf / internal nodes
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Mirroring a tree
Detailed implementations will be covered in upcoming posts.
9. Diagrammatic Overview
[ Root ]
/ \
[ Child-1 ] [ Child-2 ]
/ \ \
[Leaf] [Node] [Leaf]
This hierarchical structure provides both clarity and efficiency in data management.
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