Chapter 4: Tree

1
Chapter 4 Tree
2
Recursion (section 1.3)

• Recursive function defined in terms of itself
• f(x) 2f(x-1) x2 and f(0) 0
• f(4) 2f(3) 44
• f(3) 2f(2) 33
• f(2) 2f(1) 22
• f(1) 2f(0) 11
• f(0) 0
• f(1) 20 11 1
• f(2) 21 22 6
• f(3) 26 33 21
• f(4) 221 44 58

3
Recursion

• public static int f( int x )
• if ( x 0 ) / Base case /
• return 0
• else / General case /
• return 2 f( x-1 ) xx

4
Rules of Recursion

• Base case Always have some base cases, which
  can be solved without recursion
• Making progress Recursive call must always make
  progress toward a base case.

5
Example
Factorial f(n) n (n-1) (n-2)
1 public static long factorial ( int n )
if ( n lt 1) return 1 else
return n factorial( n - 1 )
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Example
/ print integer digit by digit / public static
void printOut( int n ) if( n gt 10 )
printOut( n / 10 ) printDigit( n 10
) printOut(123) calls printOut(12)
printOut(12) calls printOut(1)
printOut(1) calls printDigit(1) printOut(12)
calls printDigit(2) printOut(123) calls
printDigit(3)
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Example
/ print list / public static void printList(
) if ( isEmpty( ) )
System.out.println(Empty list) else
printList( header.next ) private void
printList( ListNode n ) if ( n ! null )
System.out.println(n.element)
printList(n.next)
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Bad Use of Recursion

• public static int bad( int n)
• if( n 0)
• return 0
• else
• return bad( n/31 ) n -1
• bad(1) calls bad(1) ---gt not progress to the base
  case
• bad(2) cannot be evaluated either since it also
  calls bad(1)
• bad(3), bad(4), bad(5) call bad(2)
• This program doesnt work for any value of n,
  except 0.

9
Horrible Use of Recursion
/ Compute fibonaci numbers 1,1,2,3,5,8,13,.../ p
ublic static long fib ( int n ) if ( n lt
1) / fib(0) fib(1) 1 / return 1
else return fib( n - 1 ) fib( n - 2
) fib(n-1) fib( n - 1 - 1) fib( n - 1 -
2) fib(n-1) recomputes fib( n - 2 ) So, the
running time grows exponentially
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From recursion to loop

• / Compute fibonaci numbers 1,1,2,3,5,8,13,.../
• public static long fib ( int n )
• int a1, b1, next
• if ( n lt 1 ) return 1
• for ( i2 iltn i)
• next ab
• a b
• b next
• return next

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Tree
12
Anatomy of Tree
root
edge, branch
nodes
depth
depth0 at root
parent
child
path
height
height0 at leaves
siblings
log
leaves
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Tree A Recursive Structure
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Tree
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Implementation of Trees
class TreeNode Object element
TreeNode firstChild TreeNode nextSibling
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Preorder Traversal
A B E F C D G H I
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Postorder Traversal
E F B C H I G D A
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Directory Tree
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Listing a directory (preorder traversal)
private void listAll( int depth )
printName( depth ) if ( isDirectory() )
for each file c in this directory (for each
child) c.listAll( depth 1
) public void listAll() listAll( 0 )
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Preorder listing of the directory
/home mark book ch1
ch2 ch3 junk alex
junk work
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Calculate the size of a directory (postorder
traversal)
public int size() int totalSize
sizeOfThisFile() if ( isDirectory() )
for each file c in this directory (for each
child) totalSize c.size()
return totalSize
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Trace of the size function
ch1 (3) 3 ch2 (2) 2
ch3 (4) 4 book (1) 10
junk (6) 6 mark (1) 17 junk
(8) 8 work (1) 1 alex
(1) 10 /home (1) 28
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Binary Tree
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Binary Tree
class BinaryNode Object element
BinaryNode left BinaryNode right
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Binary Tree
Balanced Binary Tree
Worse Case
number of nodes 2h1 - 1 height log(n1) - 1
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Inorder Traversal
D B E A C G F H
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Expression Tree
(a(bc))(((de)f )g) abcdefg