Computer Science and Software Engineering University of Wisconsin - Platteville 10. Binary Search Tree Yan Shi CS/SE 2630 Lecture Notes Partially adopted from C++ Plus Data Structure textbook slides

Computer Science and Software Engineering University of Wisconsin - Platteville 10. Binary Search Tree Yan Shi CS/SE 2630 Lecture Notes Partially adopted from C++ Plus Data Structure textbook slides

Review: Binary Search in a Sorted Array Based List Examines the element in the middle of the array. Is it the sought item? If so, stop searching. Is the middle element too small? Then start looking in second half of array. Is the middle element too large? Then begin looking in first half of the array. Repeat the process in the half of the list that should be examined next. Stop when item is found, or when there is nowhere else to look and item has not been found.

Motivations for Binary Search Tree array-based list: — fast search: binary search O(log2N) — slow insertion/deletion linear linked list: — fast insertion, deletion — slow search Can we have fast search, addition and deletion? — Binary search tree

Binary Tree A binary tree is a structure in which: — Each node can have at most two children, and in which a unique path exists from the root to every other node. — The two children of a node are called the left child and the right child, if they exist.

Binary Search Tree (BST) A special kind of binary tree in which: — Each node contains a distinct data value — The key values in the tree can be compared using “greater than” and “less than”, and — The key value of each node in the tree is less than every key value in its right sub-tree, and greater than every key value in its left sub-tree. Similar as sorted list, the order is maintained when inserting new items into the tree

Example Insert ‘J’ ‘E’ ‘F’ ‘T’ ‘A’ to an empty BST. ‘J’ ‘T’ ‘E’ ‘A’ ‘F’ Insert ‘A’ ‘E’ ‘F’ ‘J’ ‘T’ to an empty BST. The shape of a BST depends on • its key values and • their order of insertion ‘A’ ‘E’ ‘F’ ‘J’ ‘T’

Count Nodes in a BST CountNodes Algorithm v1: if left(tree) is NULL AND right(tree) is NULL return 1 else return CountNodes(Left(tree)) + CountNodes(Right(tree)) + 1 What if Left(tree) is NULL?

Count Nodes in a BST CountNodes Algorithm v2: if (Left(tree) is NULL) AND (Right(tree) is NULL) return 1 else if Left(tree) is NULL return CountNodes(Right(tree)) + 1 else if Right(tree) is NULL return CountNodes(Left(tree)) + 1 else return CountNodes(Left(tree)) + CountNodes(Right(tree)) + 1 What if the initial tree is NULL?

Count Nodes in a BST CountNodes Algorithm v3: if tree is NULL return 0 else if (Left(tree) is NULL) AND (Right(tree) is NULL) return 1 else if Left(tree) is NULL return CountNodes(Right(tree)) + 1 else if Right(tree) is NULL return CountNodes(Left(tree)) + 1 else return CountNodes(Left(tree)) + CountNodes(Right(tree)) + 1 Can we simplify it somehow?

Count Nodes in a BST CountNodes Algorithm Final Version: if tree is NULL return 0 else return CountNodes(Left(tree)) + CountNodes(Right(tree)) + 1 Does it cover all possible situations?

Search an item in a BST Search Algorithm Given a BST tree and an item to find: if tree is NULL found = false else if ( item < current node’s info ) Search( left(tree), item ) else if ( item > current node’s info ) Search( right(tree), item ) else found = true Similar algorithm for retrieving an item given its key

Insert an item to a BST Use recursion Base case: — tree is NULL (empty tree or leaf node’s left/right) — CANNOT add item already in the BST! General case: — if item < tree->info, Insert item to left(tree) — else if item > tree->info, Insert item to right(tree)

The “tree” parameter Treat “tree” as a pointer pointing within the tree! What happens when creating a new node? — tree become a real address instead of NULL! Make “tree” a reference parameter!

Delete an item in the BST Use recursion Base case: — If item's key matches key in Info(tree), delete node pointed to by tree. — If tree is empty, cannot delete. General case: — if item < tree->info, Delete item in left(tree) — else if item > tree->info, Delete item in right(tree) How to delete a node?

Code for Delete bool BST::Delete( TNode * & tree, Type item ) { we are changing the structure of the tree! if ( tree == NULL ) return false; if ( item < tree->info ) return Delete( tree->left, item ); else if ( item > tree->info ) return Delete( tree->right, item ); else { // How to do this? DeleteNode( tree ); return true; } }

Delete a Node in BST Algorithm: if (Left(tree) is NULL) AND (Right(tree) is NULL) Set tree to NULL else if Left(tree) is NULL Set tree to Right(tree) else if Right(tree) is NULL Set tree to Left(tree) else Find predecessor Set Info(tree) to Info(predecessor) Delete predecessor Indirect Recursion!

How to find a predecessor? Algorithm: if ( tree->left != NULL ) { tree = tree->left; while (tree->right != NULL) tree = tree->right; predecessor = tree->info; }

Tree Traversal In-order — from smallest to largest pre-order — current info first — then left — then right post-order — left first — then right — then current info

In-Order Print Algorithm: if tree is not NULL InOrderPrint( left(tree) ) print tree->info InOrderPrint( right(tree) ) How about pre-order and post-order print?

Big-3 for BST Destructor: — need to delete all node — traverse the tree in which order? post-order! Copy constructor and assignment operator: — How to copy a tree? — traverse the tree in which order? — How to implement operator=? pre-order!

Iterative Solution for Search Algorithm for search: Has( item ) Set nodePtr to tree while more elements to search if item < Info(nodePtr) Set nodePtr to Left(nodePtr) else if item > Info(nodePtr) Set nodePtr to Right(nodePtr) else return true return false

Iterative Solution for Insert 1. create a node to contain the new item 2. find the insertion place — similar as the previous search — need to keep track of parent as well 3. insert new node — insert as either the left or right child of parent

Full and Complete BST Full Binary Tree: A binary tree in which all of the leaves are on the same level and every non-leaf node has two children Complete Binary Tree: A binary tree that is either full or full through the next-to-last level, with the leaves on the last level as far to the left as possible

Array Representation of BST For any tree node nodes[index] • left child: nodes[index*2 + 1] • right child: nodes[index*2 + 2] • parent: nodes[(index – 1)/2]