Data Structures and Algorithms Lecture: Binary Trees, Heaps, and Binary Search Trees, Lecture notes of Data Structures and Algorithms

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Data Structures and

Algorithms

LECTURE 09: BINARY TREES, HEAPS, BINARY SEARCH TREES

• Binary Trees
  • Traversal algorithms
• Heaps
  • Binary heap, Min/Max heaps
• Binary Search Trees

Contents

2

• ADS representing tree like hierarchy
• Each node has at most two children
  • Children are called left and right
  • The parent is also called source

Binary Tree

S L R

• Binary trees : the most widespread form
  • Each node has at most 2 children ( left and right )

Binary Trees

5 17 9 15 6 5 8 Root node Left subtree Right child Left child

• Complete – nodes are filled top to bottom and left to right

Types of Binary Trees

7 17 9 15 6 5 8 (^13 ) 23

• Perfect – combines complete and full
  • leafs are at the same level , other nodes have two children

Types of Binary Trees

8 17 9 15 6 5 8 13 42 23 (^7 3 1 31 96 )

• Fields and constructor: Solution: BT Traversals - Constructor 10 public class BinaryTree implements AbstractBinaryTree { private E key; private BinaryTree left; private BinaryTree right; public BinaryTree(E key, BinaryTree left, BinaryTree right) { this.key = key; this.left = left; this.right = right; }

Solution: BT Traversals - Print 11 Process Node Traverse Left Traverse Right public String asIndentedPreOrder(int indent) { String out = createPadding(indent) + getKey(); if (getLeft() != null) { out +="\n" + getLeft().asIndentedPreOrder(indent + 2); } if (getRight() != null) { out +="\n" + getRight().asIndentedPreOrder(indent + 2); } return out; }

• Left  Root  Right Binary Trees Traversal: In-order 17 9 25 (^31120 )  3 9 11 17 20 25 31 inOrder (node) { if (node != null) { inOrder(node.left) print node.key inOrder(node.right) } }

• Left  Right  Root Binary Trees Traversal: Post-order 17 9 25 (^31120 )  3 11 9 20 31 25 17 postOrder (node) { if (node != null) } postOrder(node.left) postOrder(node.right) print node.key } }

Heaps

Heap, Binary Heap

• Heap
  • Tree-based data structure
  • Stored in an array
• Heaps hold the heap property for each node:
  • Min Heap
    • parent ≤ children
  • Max Heap
    • parent ≥ children

What is Heap?

• Binary heap can be efficiently stored in an array
• Parent(i) = (i - 1) / 2
• Left(i) = 2 * i + 1; Right(i) = 2 * i + 2 Binary Heap – Array Implementation 19 heap and shape properties are satisfied 5 4 1 (^3 ) 5 4 1 3 2

• To preserve heap properties :
  • Insert at the end
  • Heapify element up
• Right: Max Heap
  • Insert 16
  • Insert 25

Heap Insertion

20 17 9 15 (^6 5 8 ) Satisfy heap property Promote while element > parent 25