Binary Trees

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Chapter 11

• Binary Trees

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Chapter Objectives

• Learn about binary trees
• Explore various binary tree traversal algorithms
• Learn how to organize data in a binary search
  tree
• Discover how to insert and delete items in a
  binary search tree
• Explore nonrecursive binary tree traversal
  algorithms
• Learn about AVL (height-balanced) trees

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Binary Trees

• Definition A binary tree, T, is either empty or
  such that
• T has a special node called the root node
• T has two sets of nodes, LT and RT, called the
  left subtree and right subtree of T,
  respectively
• LT and RT are binary trees

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Binary Tree
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Binary Tree With One Node
The root node of the binary tree A LA
empty RA empty
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Binary Trees With Two Nodes
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Binary Trees With Two Nodes
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Various Binary Trees With Three Nodes
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Binary Trees

• Following struct defines the node of a binary
  tree
• templateltclass elemTypegt
• struct nodeType
• elemType info
• nodeTypeltelemTypegt llink
• nodeTypeltelemTypegt rlink

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Nodes

• For each node
• Data is stored in info
• The pointer to the left child is stored in llink
• The pointer to the right child is stored in rlink

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General Binary Tree
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Binary Tree Definitions

• Leaf node that has no left and right children
• Parent node with at least one child node
• Level of a node number of branches on the path
  from root to node
• Height of a binary tree number of nodes no the
  longest path from root to node

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Height of a Binary Tree

• Recursive algorithm to find height of binary
  tree
• (height(p) denotes height of binary tree with
  root p)
• if(p is NULL)
• height(p) 0
• else
• height(p) 1 max(height(p-gtllink),
  height(p-gtrlink))

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Height of a Binary Tree

• Function to implement above algorithm
• templateltclass elemTypegt
• int height(nodeTypeltelemTypegt p)
• if(p NULL)
• return 0
• else
• return 1 max(height(p-gtllink),
• height(p-gtrlink))

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Copy Tree

• Useful operation on binary trees is to make
  identical copy of binary tree
• Use function copyTree when we overload assignment
  operator and implement copy constructor

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Copy Tree

• templateltclass elemTypegt
• void copyTree(nodeTypeltelemTypegt
  copiedTreeRoot,
• nodeTypeltelemTypegt otherTreeRoot)
• if(otherTreeRoot NULL)
• copiedTreeRoot NULL
• else
• copiedTreeRoot new nodeTypeltelemTypegt
• copiedTreeRoot-gtinfo otherTreeRoot-gtinfo
  
• copyTree(copiedTreeRoot-gtllink,
  otherTreeRoot-gtllink)
• copyTree(copiedTreeRoot-gtrlink,
  otherTreeRoot-gtrlink)
• //end copyTree

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Binary Tree Traversal

• Must start with the root, then
• Visit the node first or
• Visit the subtrees first
• Three different traversals
• Inorder
• Preorder
• Postorder

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Traversals

• Inorder
• Traverse the left subtree
• Visit the node
• Traverse the right subtree
• Preorder
• Visit the node
• Traverse the left subtree
• Traverse the right subtree

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Traversals

• Postorder
• Traverse the left subtree
• Traverse the right subtree
• Visit the node

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Binary Tree Inorder Traversal
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Binary Tree Inorder Traversal
templateltclass elemTypegt void inorder(nodeTypeltele
mTypegt p) if(p ! NULL)
inorder(p-gtllink) coutltltp-gtinfoltlt
inorder(p-gtrlink)
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Binary Tree Traversals
templateltclass elemTypegt void postorder(nodeTypelte
lemTypegt p) if(p ! NULL)
postorder(p-gtllink) postorder(p-gtrlink)
coutltltp-gtinfoltlt 1
templateltclass elemTypegt void preorder(nodeTypeltel
emTypegt p) if(p ! NULL)
coutltltp-gtinfoltlt preorder(p-gtllink)
preorder(p-gtrlink)
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Implementing Binary Trees class binaryTreeType
Functions

• Public
• isEmpty
• inorderTraversal
• preorderTraversal
• postorderTraversal
• treeHeight
• treeNodeCount
• treeLeavesCount
• destroyTree

• Private
• copyTree
• Destroy
• Inorder, preorder, postorder
• Height
• Max
• nodeCount
• leavesCount

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Binary Search Trees

• Data in each node
• Larger than the data in its left child
• Smaller than the data in its right child
• A binary search tree,t, is either empty or
• T has a special node called the root node
• T has two sets of nodes, LT and RT, called the
  left subtree and right subtree of T, respectively
• Key in root node larger than every key in left
  subtree and smaller than every key in right
  subtree
• LT and RT are binary search trees

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Binary Search Trees
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Operations Performed on Binary Search Trees

• Determine whether the binary search tree is empty
• Search the binary search tree for a particular
  item
• Insert an item in the binary search tree
• Delete an item from the binary search tree

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Operations Performed on Binary Search Trees

• Find the height of the binary search tree
• Find the number of nodes in the binary search
  tree
• Find the number of leaves in the binary search
  tree
• Traverse the binary search tree
• Copy the binary search tree

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Binary Search Tree Analysis
Worst Case Linear tree
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Binary Search Tree Analysis

• Theorem Let T be a binary search tree with n
  nodes, where n gt 0.The average number of nodes
  visited in a search of T is approximately
  1.39log2n
• Number of comparisons required to determine
  whether x is in T is one more than the number of
  comparisons required to insert x in T
• Number of comparisons required to insert x in T
  same as the number of comparisons made in
  unsuccessful search, reflecting that x is not in T

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Binary Search Tree Analysis

• It follows that

It is also known that
Solving Equations (11-1) and (11-2)
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Nonrecursive Inorder Traversal
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Nonrecursive Inorder Traversal General Algorithm

• current root //start traversing the binary
  tree at
• // the root node
• while(current is not NULL or stack is nonempty)
• if(current is not NULL)
• push current onto stack
• current current-gtllink
• else
• pop stack into current
• visit current //visit the node
• current current-gtrlink //move to
  the
• //right
  child

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Nonrecursive Preorder Traversal General Algorithm

• 1. current root //start the traversal at the
  root node
• 2. while(current is not NULL or stack is
  nonempty)
• if(current is not NULL)
• visit current
• push current onto stack
• current current-gtllink
• else
• pop stack into current
• current current-gtrlink //prepare to
  visit
• //the right
  subtree

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Nonrecursive Postorder Traversal

• current root //start traversal at root node
• v 0
• if(current is NULL)
• the binary tree is empty
• if(current is not NULL)
• push current into stack
• push 1 onto stack
• current current-gtllink
• while(stack is not empty)
• if(current is not NULL and v is 0)
• push current and 1 onto stack
• current current-gtllink

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Nonrecursive Postorder Traversal (Continued)

• else
• pop stack into current and v
• if(v 1)
• push current and 2 onto stack
• current current-gtrlink
• v 0
• else
• visit current

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AVL (Height-balanced Trees)

• A perfectly balanced binary tree is a binary tree
  such that
• The height of the left and right subtrees of the
  root are equal
• The left and right subtrees of the root are
  perfectly balanced binary trees

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Perfectly Balanced Binary Tree
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AVL (Height-balanced Trees)

• An AVL tree (or height-balanced tree) is a binary
  search tree such that
• The height of the left and right subtrees of the
  root differ by at most 1
• The left and right subtrees of the root are AVL
  trees

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AVL Trees
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Non-AVL Trees
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Insertion Into AVL Tree
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Insertion Into AVL Trees
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Insertion Into AVL Trees
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Insertion Into AVL Trees
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Insertion Into AVL Trees
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AVL Tree Rotations

• Reconstruction procedure rotating tree
• left rotation and right rotation
• Suppose that the rotation occurs at node x
• Left rotation certain nodes from the right
  subtree of x move to its left subtree the root
  of the right subtree of x becomes the new root of
  the reconstructed subtree
• Right rotation at x certain nodes from the left
  subtree of x move to its right subtree the root
  of the left subtree of x becomes the new root of
  the reconstructed subtree

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AVL Tree Rotations
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AVL Tree Rotations
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AVL Tree Rotations
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AVL Tree Rotations
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AVL Tree Rotations
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AVL Tree Rotations
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Deletion From AVL Trees

• Case 1 the node to be deleted is a leaf
• Case 2 the node to be deleted has no right
  child, that is, its right subtree is empty
• Case 3 the node to be deleted has no left child,
  that is, its left subtree is empty
• Case 4 the node to be deleted has a left child
  and a right child

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Analysis AVL Trees
Consider all the possible AVL trees of height h.
Let Th be an AVL tree of height h such that Th
has the fewest number of nodes. Let Thl denote
the left subtree of Th and Thr denote the right
subtree of Th. Then
where Th denotes the number of nodes in Th.
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Analysis AVL Trees
Suppose that Thl is of height h 1 and Thr is of
height h 2. Thl is an AVL tree of height h 1
such that Thl has the fewest number of nodes
among all AVL trees of height h 1. Thr is an
AVL tree of height h 2 that has the fewest
number of nodes among all AVL trees of height h
2. Thl is of the form Th -1 and Thr is of the
form Th -2. Hence
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Analysis AVL Trees
Let Fh2 Th 1. Then
Called a Fibonacci sequence solution to Fh is
given by
Hence
From this it can be concluded that
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Chapter Summary

• Binary trees
• Binary search trees
• Recursive traversal algorithms
• Nonrecursive traversal algorithms
• AVL trees