TreesTypes and Operations

1. General Trees2. Binary Search Trees3. AVL Trees4. Heap Trees

Agenda

Insertion◦FIFO◦LIFO◦Key-sequenced Insertion

DeletionChanging a General Tree into a Binary Tree

1. General Trees

Given the parent Node a new node may be inserted as

FIFO

Insertion

Data Structures: A Pseudocode Approach with C 5

First in-first out (FIFO) insertion

Given the parent Node a new node may be inserted as

FIFO LIFO

Insertion

Data Structures: A Pseudocode Approach with C 7

Last in-first out (LIFO) insertion

Given the parent Node a new node may be inserted as

FIFO LIFO Key-sequenced Insertion

Insertion

Data Structures: A Pseudocode Approach with C 9

Key-sequenced insertion

Insertion◦FIFO◦LIFO◦Key-sequenced Insertion

DeletionChanging a General Tree into a Binary Tree

1. General Trees

For general trees nodes to be deleted are restricted to be “leaves”

Otherwise a node maybe “purged”, i.e. a node is deleted along with all its children

Deletion

Insertion◦FIFO◦LIFO◦Key-sequenced Insertion

DeletionChanging a General Tree into a Binary Tree

1. General Trees

Changing the meaning of the two pointers:

Leftchild …..first childRightchild ….. Next siblings

Changing into Binary Trees

Data Structures: A Pseudocode Approach with C 14

Changing a General Tree to a Binary Tree

Data Structures: A Pseudocode Approach with C 15

Changing a General Tree to a Binary Tree

Data Structures: A Pseudocode Approach with C 16

Changing a General Tree to a Binary Tree

1. General Trees2. Binary Search Trees3. AVL Trees4. Heap Trees

Agenda

Basic ConceptsBST OperationsThreaded Trees

2. Binary Search Tree

All items in left subtree < rootAll items in right subtree > root

Basic Concepts

Binary Search Trees

A binary search tree Not a binary search tree

Binary Search TreeTwo binary search trees representing the same set:

Basic ConceptsBST OperationsThreaded Trees

2. Binary Search Tree

TraversalSearch

◦Smallest ……….. ?◦Largest …………?◦Specific element

InsertionDeletion

BST Operations

Inorder traversal of BST Print out all the keys in sorted order

Inorder: 2, 3, 4, 6, 7, 9, 13, 15, 17, 18, 20

TraversalSearch

◦Smallest ……….. ?◦Largest …………?◦Specific element

InsertionDeletion

BST Operations

findMin/ findMax Return the node containing the smallest element in the tree

Start at the root and goes left/right as long as there is a left/right child. The stopping point is the smallest/largest element

Time complexity = O(height of the tree)

Searching BST (specific elem) If we are searching for 15, then we are

done. If we are searching for a key < 15, then we

should search in the left subtree. If we are searching for

a key > 15, then we should search in the right subtree.

Searching BST

TraversalSearch

◦Smallest ……….. ?◦Largest …………?◦Specific element

InsertionDeletion

BST Operations

insert Proceed down the tree as you would with a find If X is found, do nothing (or update something) Otherwise, insert X at the last spot on the path traversed

Time complexity = O(height of the tree)

TraversalSearch

◦Smallest ……….. ?◦Largest …………?◦Specific element

InsertionDeletion

BST Operations

delete When we delete a node, we need to

consider how we take care of the children of the deleted node.

This has to be done such that the property of the search tree is maintained.

deleteThree cases:(1) the node is a leaf

◦ Delete it immediately

(2) the node has one sub-tree (right or left)◦ Adjust a pointer from the parent to bypass that node

delete(3) the node has 2 children

◦ replace the key of that node with the minimum element at the right subtree (or the maximum element at the left subtree)

◦ delete the minimum element Has either no child or only right child

because if it has a left child, that left child would be smaller and would have been chosen. So invoke case 1 or 2.

Time complexity = O(height of the tree)

delete

Basic ConceptsBST OperationsThreaded Trees

2. Binary Search Tree

Sparing recursion and stackMaking use of null right child of leaves to point to next node

Threaded Trees

1. General Trees2. Binary Search Trees3. AVL Trees4. Heap Trees

Agenda

PropertiesOperations

3. AVL Trees

It is a balanced binary tree (definition of Russian mathematicians Adelson-Velskii and Landis)

The height of its sub-trees differs by no more than one (its balance factor is -1, 0, or 1), and its subtrees are also balanced.

Properties of AVL Trees

A sub tree is called Left high (LH) if its balance is 1Equally high (EH) if it is 0Right high (RH) if it is -1

Properties of AVL Trees

Insertion and deletion are same as in BST

If unbalance occurs corresponding rotations must be performed to restore balance

Operations on AVL Trees

Balanced trees: AVL tree rotations Steps:

◦ Check if case is case 1 or 2 of the following and act accordingly

◦Case 1: tree is left high & out-of-balance is created by a adding node to the

left of the left sub-tree

◦ …… One right rotation is needed Rotate out-of-balance node right

Case 1: single R-rotation

h

h hhh

h+1h+2 h+

1

h+1

•Case 1 * Tree is left balanced unbalance is caused by node on the left of left sub-tree

Balanced trees: AVL tree rotations

◦Case 2: tree is left highout-of-balance is created by a adding node to

the right of the left sub-tree

◦ …… Two rotations are needed:Move from bottom of left sub-tree upwards

till an unbalanced node is found and rotate it left

Rotate left sub-tree right

Case 2: Double LR-rotation

h+2h+1

Add node to right of left balanced subtree

First rotation .. Left rotation of unbalanced node c

Second rotation … Right rotation of left sub-tree g

h

h

h h

hh+1

h+2

h+2h+

1h+1

End