1. Binary Trees

2. Binary Trees Trees with no more than two children per node
We call the children of a node left child and right child
11/29/2018

3. Binary Trees
Root
Left child
Right child
Right Sub-tree
11/29/2018

4. Recursive Definition of Binary Trees
A binary tree, T, is either empty or such that:
T has a root node;
T has two sets of nodes, LT and RT, called the left sub-tree and right sub-tree of T, respectively;
LT and RT are binary trees
Left Sub-tree
Right Sub-tree
11/29/2018

5. Full Binary Tree Every leaf node has the same depth
Every non-leaf node has two children
Every non-leaf node has two children
Every leaf node has the same depth
11/29/2018

6. Full Binary Tree Not full Not full Non-leaf node with only one child
Leaf nodes with different depths
Not full
Not full
Non-leaf node with only one child
11/29/2018

7. Number of Nodes in a Full Binary Tree
Depth
Nodes at Depth
Total Nodes 1 2 3 4 n 1
N0=1 2
N1=1+2=3 4
N2=3+4=7 8
N3=7+8=15 16
N4=15+16=31
Nn=Nn-1+2n
=2n+1-1 2n
11/29/2018

8. Only one place to add another node
Complete Binary Tree
Every level but lowest has all possible nodes
If lowest level is not full, all nodes as far left as possible
Only one place to add another node
11/29/2018

9. Binary Tree Applications
Expression trees
Decision Trees
Huffman codes
Search trees (next topic)
11/29/2018

10. Arithmetic Expression Tree
Binary tree associated with an arithmetic expression
non-leaf nodes: operators
leaf nodes: operands
Example: arithmetic expression tree for the expression (2  (a - 1)) + (3  b) +
2  (a - 1)
3  b   3 b 2
a-1 a 1 -
11/29/2018

11. Decision Tree Binary tree associated with a decision process
non-leaf nodes: questions with yes/no answer
leaf nodes: decisions
Example: mail service decision
11/29/2018

12. Decision Tree YES NO YES NO NO YES Is speed of delivery important?
Is contents valuable?
YES NO
Is contents fragile? NO
YES
Insured air freight
Air freight
Rail freight
Bus parcel express
11/29/2018

13. Huffman codes Used for signal compression
Suppose we have a message consisting of 5 symbols, e.g. [►♣♣♠☻►♣☼►☻]
How can we code this message using 0/1 so the coded message will have minimum length (for transmission or saving!)
5 symbols  at least 3 bits (fixed length codes)
For a simple encoding,
length of code is 10*3=30 bits
11/29/2018

14. Huffman codes Uses variable length codes
Intuition: Those symbols that are more frequent should have smaller codes
For Huffman code,
length of encoded message
will be ►♣♣♠☻►♣☼►☻
=3*2 +3*2+2*2+3+3=24bits
11/29/2018

15. Huffman Coding Algorithm
1. Take the two least probable symbols:
Assign them longest codes, equal length, differing in last digit
2. Combine these two symbols into a single Virtual symbol, and repeat.
11/29/2018

16. Huffman Coding Algorithm
Symbols/Probability ={ a/0.25 , b/0.25 , c/0.2 , d/0.15 , e/0.15 }
1.0 1
0.55 1
0.45 1
0.3 1 a
0.25 b
0.25 c
0.2 d
0.15 e
0.15 00 10 11
010
011
11/29/2018

17. Binary Tree Representations
There are two ways to represent a Tree:
Arrays
Linked nodes
11/29/2018

18. Array Representation of Binary Trees
Number the nodes as follows:
Number the nodes 1 through 2h – 1.
Number by levels from top to bottom.
Within a level number from left to right. 1 2 3 4 6 5 7 8 9
11/29/2018 10 11 12 13 14 15

19. Array Representation of Binary Trees
Parent of node i is node i / 2, unless i = 1.
Node 1 is the root and has no parent.
Left child of node i is node 2i, unless 2i > n, where n is the number of nodes.
If 2i > n, node i has no left child.
Right child of node i is node 2i+1, unless 2i+1 > n, where n is the number of nodes.
If 2i+1 > n, node i has no right child.
11/29/2018

20. Array Representation of Binary Trees
Number the nodes as described earlier. The node that is numbered i is stored in tree[i]. b a c d e f g h i j 1 2 3 4 5 6 7 8 9 10
tree[] 5 10 a b c d e f g h i j
11/29/2018

21. Array Representation of Binary Trees1 3 c 7 d 15
tree[] 5 10 a - b c 15 d
An n node binary tree needs an array whose length is between n+1 and 2n.
11/29/2018

22. Linked Representation of Binary Trees
Each binary tree node is represented as an object whose data type is BTNode.
class BTNode {
Object data; // any type
BTNode left; // left sub-Tree
BTNode right; //right sub-Tree }
11/29/2018

23. Linked Representation of Binary Trees
class BTNode {
Object data;
// any type
BTNode left;
// left sub-Tree
BTNode right;
//right sub-Tree }
11/29/2018

24. Generic Binary node structures
Use a generic type parameter for the data
class BTNode<E> {
E data;
BTNode left;
BTNode right; }
11/29/2018

25. BTNode class (assuming Object data)
Constructor
Public BTNode(
Object initialData,
BTNode initialLeft,
BTNode initialRight ) {
data = initialData;
left = initialLeft;
right = initialRight; }
11/29/2018

26. BTNode class (assuming Object data)
Methods
regular accessors and manipulators
e.g., public BTNode getLeft() //Gets reference to left child {
return left; }
public void setData(Object newData)
//Set data of this node
data = newData;
11/29/2018

27. BTNode class (assuming Object data)
Methods
properties of node in tree
e.g., public boolean isLeaf() {
return (left == null) && (right == null); }
11/29/2018

28. BTNode class (assuming Object data)
Methods
tree methods
e.g., public void preOrderPrint()
public BTNode removeLeftmost()
static tree methods
e.g., public static treeCopy(BTNode source)
public static int treeSize(BTNode root)
public static int height() // not in Main
11/29/2018

29. public BTNode removeLeftmost()
starting from root, how do you find the leftmost node?
how do you remove the leftmost node and repair the tree structure?
11/29/2018

30. public BTNode removeLeftmost()
How do you find the leftmost node?
follow left child links to node with no left child
11/29/2018

31. public BTNode removeLeftmost()
How do you remove the leftmost node and repair the tree structure?
leftmost node has no left child
attach right child (may be null) to parent of leftmost node as left child
11/29/2018

32. public BTNode removeLeftmost(){
if (left==null)
return right;
left = left.removeLeftmost();
return this; }
11/29/2018

33. Traversals of Binary Trees
Pre-order
In-order
Post-order
11/29/2018

34. Traversals of Binary Trees
Pre-order
Process the root
Process the nodes in the left sub-Tree with a recursive call
Process the nodes in the right sub-Tree with a recursive call 1 2 3
11/29/2018

35. Traversals of Binary Trees
Pre-order
public void printPreOrder() {
System.out.println(data); //1
if (left != null)
left.printPreOrder(); //2
if (right != null)
right.printPreOrder(); //3 } 1 2 3
11/29/2018

36. Traversals of Binary Trees
In-order
Process the nodes in the left sub-Tree with a recursive call
Process the root
Process the nodes in the right sub-Tree with a recursive call 1 3 2
11/29/2018

37. Traversals of Binary Trees
In-order
public void printInOrder() {
if (left != null)
left.printInOrder(); //1
System.out.println(data); //2
if (right != null)
right.printInOrder(); //3 } 1 3 2
11/29/2018

38. Traversals of Binary Trees
Post-order
Process the nodes in the left sub-Tree with a recursive call
Process the nodes in the right sub-Tree with a recursive call
Process the root 1 2 3
11/29/2018

39. Traversals of Binary Trees
Post-order
public void printPostOrder() {
if (left != null)
left.printPostOrder(); //1
if (right != null)
right.printInOrder(); //2
System.out.println(data); //3 } 1 2 3
11/29/2018

40. Recursive method: height
calculating height of a tree of BTNode: (=height of root)
height: length of path to deepest descendent leaf
Recursive design:
BASE CASES:
height of empty tree is -1
height of one-node tree is 0
RECURSIVE CASES:
height is 1 more than height of higher child 3 2 1

41. Recursive view of the node height calculation: HT = Max (HL + 1, HR + 1)
11/29/2018
Data Structures & Problem Solving using JAVA/2E Mark Allen Weiss © Addison Wesley

42. Recursive method: height
calculating height of a tree of BTNode: (=height of root)
public static int height(BTNode b) {
if (b == null)
return -1;
return 1 + Math.max(height(b.left),height(b.right)); }

43. Recursive view used to calculate the size, ST, of a tree: ST = SL + SR + 1.
11/29/2018
Data Structures & Problem Solving using JAVA/2E Mark Allen Weiss © Addison Wesley

44. Recursive method: size of tree
How many nodes in a tree?
write recursive method with header:
static int size (BTNode b) {
if (b == null )
return 0;
return 1 + size(b.left) + size(b.right); }