Binary search tree and its explaination, Slides of Data Structures and Algorithms

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Data Structures and Algorithms

LECTURE 14 – BINARY TREES

General / Binary Tree

• General tree – a node can have any number of children
• Binary tree – a node can have at most two children

Binary Search – (Search 13)

BINARY TREE

Binary Search Tree

• A fundamental data structure in computer science
• Used for rapidly storing and retrieving data
• Composed of nodes which store data and links to up to two other child nodes
• The nodes on the left have a lesser key values (i.e., the value used to search for a Node

in the tree), and the nodes on the right have a higher key value

• Corollary:
  • Leaves on the farthest left of the tree have the lowest values
  • Leaves on the right of the tree have the greatest values

Why?

• Arrays:
  • Arrays can achieve efficient searching, e.g., binary search
  • However, inefficient insertion and deletion algorithms that require shifting data in the array
• Linked Lists
  • Efficient insertions and deletions
  • However, search algorithms are sequential which are inefficient
• Binary Search Tree
  • Efficient search, insertion, and deletion

Examples

A Binary Search Tree

Examples

A Binary Search Tree What about this one?

Examples

• None of them is a Binary Search Tree.

BST Complexity

• Number of nodes in the above given “complete” BST = n
• Height of BST = h
• Given the of height h, the value of n, i.e., all nodes in the tree are:
  • n = 2^0 + 2^1 + 2^2 + 2^3 + 2^4 …. + 2h

BST Implementation – Node

• See Handout for Source Code – “BST” //BST = Binary Search Tree class BSTNode { public : int key; BSTNode left, right; BSTNode( int el = 0, BSTNode l = NULL, BSTNode r = NULL) { key = el; left = l; right = r; } };

BST Implementation - Class class BST { public : BSTNode* root; BST(): root(NULL) {} //destroy routines ~BST(); void clear(BSTNode* ptr); //insertion routines void insert( int A); void insert_node( int B, BSTNode* &node1); //traversal routines void visit(BSTNode* node) { cout<<" :: "<<node->key<<" :: "<<endl; } void breadth_first(); void depth_first(); void preorder(BSTNode* node); void inorder(BSTNode* node); //searching routines int search( int A); BSTNode* search_node(BSTNode* node, int B); //deletion routines void deletenode( int A); void deletebycopying(BSTNode* &node2); };

BST Operations - Insertion

• A Recursive Call
• Base case
  • The tree is empty
• General case
  • Choose the left or right subtree

BST Operations - Searching

• A Recursive Call
• Base case
  • The Node is found whose key equals the required
  • Return the Node’s reference
• General case
  • If the required key is less than the current node’s key, search Left Subtree
  • If the required key is greater than or equal to the current node’s key. search Right Subtree