1. Trees

2. Outline Binary trees Binary search trees AVL trees B trees
Applications
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Data Structure: Trees

3. Trees

4. Tree ADT A tree is a collection (may be empty) of nodes, containing:
a distinguished node called the root r,
zero or more non-empty subtrees T1, T2, …, Tk,
A directed edge from the root r to the root of each subtree.
root T1 T2 Tk …
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5. Terminologies parent root children siblings subtrees grandchildren
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6. Terminologies path ancestor length of path depth
length of path from the root
descendant
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7. Terminologies
height
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8. Tree: Implementation class TreeNode { Object element;
TreeNode firstChild;
TreeNode nextSibling; }
nextSibling=null
nextSibling=null
nextsibling
firstChild=null
firstChild=null
firstChild
nextSibling=null
firstChild=null
firstChild=null
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Data Structure: Trees

9. Binary Trees

10. Binary Trees A tree in which no node can have more than 2 children.
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Data Structure: Trees

11. Binary Tree: Implementation
Class BinaryNode {
Object element;
BinaryNode left;
BinaryNode right; } 3
node.element=3
node.left=node
node.right=node 5
node.element=5
node.left=node
node.right=null 9
node.element=9
node.left=null
node.right=null
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Data Structure: Trees

12. Binary Tree: Implementation
class BinaryNode
{ // Constructors
BinaryNode ( Comparable theElement )
{ this ( theElement, null, null); }
BinaryNode
(Comparable theElement, BinaryNode lt, BinaryNode rt)
{ element = theElement;
left = lt;
right = rt; }
// Friendly data; accessible by other package routines
Comparable element; // The data in the node
BinaryNode left; // Left child
BinaryNode right; // Right child }
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Data Structure: Trees

13. Binary Search Trees

14. Binary Search Trees Properties of a binary search tree T
T is a binary tree.
For each node n in T, whose left subtree is Tl and right subtree is Tr,
the item in each node in Tl is smaller than the item in n.
the item in each node in Tr is larger than the item in n.
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Data Structure: Trees

15. Example 8 3 11 2 4 9 12 1 6 5 7
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Data Structure: Trees

16. Binary Search Trees public class BinarySearchTree
{ public BinarySearchTree( )
{ root = null; }
public void insert( Comparable x )
{ root = insert( x, root ); }
public void remove( Comparable x )
{ root = remove( x, root ); }
public Comparable find( Comparable x )
{ return elementAt( find( x, root ) ); }
public void makeEmpty( )
{ root = null; }
...
private BinaryNode root; }
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Data Structure: Trees

17. FIND 8 Find 6 < 3 11 > 2 4 9 12 > 1 6 5 7 11/16/2018
Data Structure: Trees

18. Method find private BinaryNode find ( Comparable x, BinaryNode t ) {
if( t == null ) return null;
if( x.compareTo( t.element ) < 0 )
return find( x, t.left );
else if( x.compareTo( t.element ) > 0 ) return find( x, t.right );
else return t; // Match }
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Data Structure: Trees

19. INSERT 10 8 Insert 10 > 3 11 < 2 4 9 12 10 1 6 5 7 11/16/2018
Data Structure: Trees

20. Method insert private BinaryNode insert ( Comparable x, BinaryNode t )
{ if( t == null )
t = new BinaryNode( x, null, null );
else if( x.compareTo( t.element ) < 0 )
t.left = insert( x, t.left );
else if( x.compareTo( t.element ) > 0 )
t.right = insert( x, t.right );
else ; // Duplicate; do nothing
return t; }
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Data Structure: Trees

21. FindMax, FindMin 8 3 11 2 4 9 12 max 1 6 min 5 7 11/16/2018
Data Structure: Trees

22. Methods findMin & findMax
private BinaryNode findMin( BinaryNode t )
{ if( t == null )
return null;
else if( t.left == null )
return t;
return findMin( t.left ); }
private BinaryNode findMax( BinaryNode t )
{ if( t != null )
while( t.right != null )
t = t.right;
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Data Structure: Trees

23. REMOVE 11
Remove 7 3 13 2 7 12 14 1 5 9 4 6 10
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24. REMOVE 11
Remove 7 3 13 2 7 12 14 1 9 10 8
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25. REMOVE 11
Remove 7 3 13 2 7 12 14 1 5 6 4
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Data Structure: Trees

26. Method Remove private BinaryNode remove(Comparable x,BinaryNode t)
{ if(t == null) return t; // Item not found;do nothing
if( x.compareTo(t.element) < 0 )
t.left = remove(x, t.left);
else if ( x.compareTo(t.element) > 0 )
t.right = remove(x, t.right);
else if (t.left!=null && t.right!=null) // 2 child
{ t.element = findMin(t.right).element;
t.right = remove(t.element, t.right); }
else
t = (t.left != null) ? t.left : t.right;
return t;
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Data Structure: Trees

27. AVL Trees

28. AVL Trees
An AVL tree is a binary search tree with a balance condition.
Balance condition
For every node in the tree, the height of the left & right subtrees can differ by at most 1.
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Data Structure: Trees

29. AVL Trees 11 11 3 7 2 1 13 14 12 5 6 4 3 13 2 7 12 14 1 5 8
not AVL tree
AVL tree
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30. Single Right Rotation k2 k1 k1 Zh k2 Xh+1 Xh+1 Yh Yh Zh 11/16/2018
Data Structure: Trees

31. Single Left Rotation k2 k1 k1 Xh k2 Yh Zh+1 Zh+1 Xh Yh 11/16/2018
Data Structure: Trees

32. What cannot be solved by single rotationk2 k1 k2 Xh k1 Zh
Yh+1 Zh Xh
Yh+1
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33. Double Rotation k3 k Xh Yh+1 Zh k3 k2 Xh Zh k1 Y’h Y’’h k3 k1 Xh Zh k2
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Data Structure: Trees

34. Double Rotation k3 k1 Xh Zh k2 Y’h Y’’h k3 k2 Xh Zh k1 Y’h Y’’h
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35. Height of AVL Tree
If N is the number of nodes in an AVL tree, the height
of the tree, h(N), is approximately 1.44 log(N+2)-.328.
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Data Structure: Trees

36. Class AvlNode Class AvlNode { AvlNode (Comparable theElement)
{ this(theElement, null, null); }
AvlNode ( Comparable theElement, AvlNode lt,rt )
{ element = theElement;
left=lt; right=rt; }
private static int height (AvlNode t)
{ return t==null ? -1 : t.height; }
...
Comparable element;
AvlNode left;
AvlNode right;
int height; }
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Data Structure: Trees

37. Method insertion private AvlNode insert(Comparable x,AvlNode t)
{ if (t == null) // insert in an empty tree
t = new AvlNode( x, null, null );
// go down the left subtree
else if (x.compareTo( t.element ) < 0)
{ t.left = insert( x, t.left );
// Does insertion violate balance condition?
if(height(t.left)-height(t.right)==2)
if(x.compareTo(t.left.element)<0)
t = rotateWithLeftChild( t );
else
t = doubleWithLeftChild( t ); }
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Data Structure: Trees

38. Method: insertion // go down the right subtree
else if(x.compareTo( t.element ) > 0)
{ t.right = insert( x, t.right );
// Does insertion violate balance condition?
if(height(t.right)-height(t.left)==2)
if(x.compareTo(t.right.element)>0)
t = rotateWithRightChild( t );
else
t = doubleWithRightChild( t ); }
; // Duplicate; do nothing
t.height=max(height(t.left),height(t.right))+1;
return t; }
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Data Structure: Trees

39. Method rotateWithLeftChild
private static AvlNode rotateWithLeftChild(AvlNode k2)
{ AvlNode k1 = k2.left;
k2.left = k1.right; k1.right = k2;
k2.height =
max(height(k2.left), height(k2.right))+1;
k1.height =
max(height(k1.left), k2.height)+1;
return k1; } k2 k1 k2 k1
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Data Structure: Trees

40. Method rotateWithRightChild
private static AvlNode
rotateWithRightChild( AvlNode k1 )
{ AvlNode k2 = k1.right;
k1.right = k2.left; k2.left = k1;
k1.height =
max(height(k1.left), height(k1.right))+1;
k2.height =
max(height(k2.left), k2.height)+1;
return k1; } k2 k1 k2 k1
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Data Structure: Trees

41. Method doubleWithLeftChild
private static AvlNode doubleWithLeftChild (AvlNode k3)
{ k3.left = rotateWithRightChild(k3.left);
return(rotateWithLeftChild(K3); } k3 k2 k1 k3 k2 k1 k3 k2 k1
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Data Structure: Trees

42. Method doubleWithRightChild
private static AvlNode doubleWithRightChild (AvlNode k3)
{ k3.right = rotateWithLeftChild(k3.right);
return(rotateWithRightChild(k3); } k3 k2 k1 k3 k2 k3 k1 k2 k1
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Data Structure: Trees

43. Binary Search Trees AVL Trees
Running Time
Binary Search Trees
AVL Trees

44. Running time: Method find
BinaryNode find(Comparable x,BinaryNode t)
{ if( t == null ) return null;
if( x.compareTo( t.element ) < 0 )
return find( x, t.left );
else if( x.compareTo( t.element ) > 0 ) return find( x, t.right );
else return t; // Match }
T(0) = c h is the height of the tree.
T(h) = T(h-1) + k’
T(h) = O(h) c
T(h-1)
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Data Structure: Trees

45. Running time: Method insert
BinaryNode insert (Comparable x, BinaryNode t)
{ if( t == null )
t = new BinaryNode( x, null, null );
else if( x.compareTo( t.element ) < 0 )
t.left = insert( x, t.left );
else if( x.compareTo( t.element ) > 0 )
t.right = insert( x, t.right );
return t; }
T(0) = c h is the height of the tree.
T(h) = T(h-1) + k’
T(h) = O(h) c
T(h-1)
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Data Structure: Trees

46. Running Time: Method findMax
private BinaryNode findMax( BinaryNode t )
{ if( t != null )
while( t.right != null )
t = t.right;
return t; }
T(h)=O(h), where h is the height of the tree.
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Data Structure: Trees

47. Running Time: Method Remove
private BinaryNode remove(Comparable x,BinaryNode t)
{ if(t == null) return t;
if( x.compareTo(t.element) < 0 )
t.left = remove(x, t.left);
else if ( x.compareTo(t.element) > 0 )
t.right = remove(x, t.right);
else if (t.left!=null && t.right!=null)
{ t.element = findMin(t.right).element;
t.right = remove(t.element, t.right); }
else t = (t.left != null) ? t.left : t.right;
return t; }
T(0) = c h is the height of the tree.
T(h) = T(h-1) + f(h) f(h) = O(h).
T(h) = O(h2) c
T(h-1)
O(h)
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Data Structure: Trees

48. Running Time: Method insertion
private AvlNode insert(Comparable x,AvlNode t)
{ if (t == null) t=new AvlNode(x,null,null);
else if (x.compareTo( t.element ) < 0)
{ t.left = insert( x, t.left );
// rotate }
else if(x.compareTo( t.element ) > 0)
{ t.right = insert( x, t.right );
// rotate }
// calculate height
return t;
T(0) = c h is the height of the tree.
T(h) = T(h-1) + k’ T(h) = O(h) c
T(h-1) k k’
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Data Structure: Trees

49. Tree Traversal
For Binary Trees

50. Inorder Traversal + (a – (b * (c / d))) + (e – f) - - a * e f / b c d
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51. Method inorder public static void inorder (BinaryNode t)
{ if ( t!=null )
{ inorder(t.left);
System.out.println(t.element);
inorder(t.right); }
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52. Preorder Traversal + + – a * b / c d – e f - - a * e f / b c d
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53. Method preorder public static void preorder (BinaryNode t)
{ if ( t!=null )
{ System.out.println(t.element);
inorder(t.left);
inorder(t.right); }
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54. Postorder Traversal + a b c d / * – e f – + - - a * e f / b c d
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55. Method postorder public static void postorder (BinaryNode t)
{ if ( t!=null )
{ inorder(t.left);
inorder(t.right);
System.out.println(t.element); }
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Data Structure: Trees

56. B Trees

57. B trees
N-ary tree
Increase the breadth of trees to decrease the height
Used for indexing of large amount of data (stored in disk)
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58. Example
4 8
83 91
60 69 1 2 5 6 7 8 9 11 12 79 80 81 82 83 85 86 90 93 95 97 98 99 54 56 57 59 60 61 62 66 67 70 71 76 77 13 14 17 19 20 21 22 26 27 28 31 35 38 44 45 49 50
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59. Properties of B Trees For an M-ary B tree:
The root has up to M children.
Non-leaf nodes store up to M-1 keys, and have between M/2 and M children, except the root.
All data items are stored at leaves.
All leaves have to same depth, and store between L/2 and L data items.
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60. Search
Search for 66
4 8
83 91
60 69 1 2 5 6 7 8 9 11 12 79 80 81 82 83 85 86 90 93 95 97 98 99 54 56 57 59 60 61 62 66 67 70 71 76 77 13 14 17 19 20 21 22 26 27 28 31 35 38 44 45 49 50
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61. Insert Insert 55 Split leave 12 52 78 4 8 83 91 19 26 37 46 60 69 1 2
4 8
83 91
60 69 1 2 5 6 7 8 9 11 12 79 80 81 82 83 85 86 90 93 95 97 98 99 54 56 57 59 60 61 62 66 67 70 71 76 77 13 14 17 19 20 21 22 26 27 28 31 35 38 44 45 49 50
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62. Insert Insert 32 Insert key 31 Split leave Insert key 31 Split node
4 8
Insert key 31
Split node
83 91
60 69 1 2 5 6 7 8 9 11 12 79 80 81 82 83 85 86 90 93 95 97 98 99 54 56 57 59 60 61 62 66 67 70 71 76 77 13 14 17 19 20 21 22 26 27 28 31 35 36 38 44 45 49 50
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