1. Chapter 7. Trees & Binary Trees
Lecture 14
Chapter 7. Trees & Binary Trees
Set ADT
Introduction to trees and binary trees
ADS2 Lecture 14

2. Pictures of trees
File System

3. Pictures of trees

4. ADS2 Lecture 14

5. Pictures of trees

6. Pictures of trees

7. Pictures of trees

8. Pictures of trees

9. Pictures of trees
unrooted

10. Pictures of trees
unrooted

11. unrooted
Pictures of trees

12. unrooted
Pictures of trees
ADS2 Lecture 14 12

13. Pictures of trees
ADS2 Lecture 14 13

14. Pictures of trees
ADS2 Lecture 14 14

15. Pictures of trees
ADS2 Lecture 14 15

16. Pictures of trees

17. Pictures of trees

18. Pictures of trees

19. Pictures of trees

20. Pictures of trees

21. Pictures of trees

22. Pictures of trees

23. Nonlinear data structure Natural way to organise data File system GUI
General Trees
Nonlinear data structure
Natural way to organise data
File system
GUI
Databases
Websites
Inheritance relation in Java classes
Books (chapters, sections, subsections, subsubsections)
“nonlinear” organisational structure
Not just “before” “after”
“above” “below” “part of”
Relationships are typically
Parent
Children
Ancestors
Descendents
Siblings
ADS2 Lecture 14

24. A tree T is a set of nodes with a parent-child relationship
Formal definition
General Trees
A tree T is a set of nodes with a parent-child relationship
Each node has one parent, apart from the root (which has no parent)
A tree T is either empty or consists of a node r,
the root of T, and a (possibly empty) set of trees
whose roots are the children of r
An edge in T is a pair of nodes (u,v)
where u is parent of v or v is parent of u
ADS2 Lecture 14

25. i e c b R d j g f h General Trees a node an edge a path a a child
a parent
an ancestor
a descendant
siblings
depth of a node
height of a node
height of a tree
a leaf
an internal node
the root
a subtree i e c b R d j g f h a X n p Z

26. i e c b R d j g f h General Trees n nodes n-1 edges a no cyclesX n p Z
n nodes
n-1 edges
no cycles
unique path from a node to root

27. How might we implement a general tree?
ADS2 Lecture 14

28. i e c b R d j g f h General Trees a p Z X n
Therefore we might implement a tree as follows
Tree<E>
Node<E> root
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Note: there could be order amongst the children
ADS2 Lecture 14

29. i e c b R d j g f h Depth General Trees a Tree<E>X n p Z
Tree<E>
Node<E> root
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Depth of a node is how far it is from the root.
depth(Node<E> node)
if (node.isRoot()) return 0;
return 1 + depth(node.parent());

30. i e c b R d j g f h Height General Trees a Tree<E>X n p Z
Tree<E>
Node<E> root
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Height of a node is
If node is a leaf then 0
Otherwise 1 + the maximum of the height of its children
height(Node<E> node)
if (node.isLeaf()) return 0;
int h = 0;
for (Node<E> child : node.children()) h = Math.max(h,height(child));
return 1 + h;

31. i e c b R d j g f h Preorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the parent then visit its children

32. i e c b R d j g f h Preorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the parent then visit its children

33. i e c b R d j g f h Preorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the parent then visit its children

34. i e c b R d j g f h Preorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the parent then visit its children

35. i e c b R d j g f h Preorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the parent then visit its children

36. i e c b R d j g f h Preorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the parent then visit its children

37. i e c b R d j g f h Preorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the parent then visit its children

38. i e c b R d j g f h Preorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the parent then visit its children

39. i e c b R d j g f h Preorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the parent then visit its children

40. i e c b R d j g f h Preorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the parent then visit its children

41. i e c b R d j g f h Preorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the parent then visit its children

42. i e c b R d j g f h Preorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the parent then visit its children

43. i e c b R d j g f h Preorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the parent then visit its children

44. i e c b R d j g f h Preorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the parent then visit its children

45. i e c b R d j g f h Preorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the parent then visit its children

46. i e c b R d j g f h Preorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the parent then visit its children
preorder(Node<E> node)
if (node != null) {
visit(node);
for (Node<E> child : node.children()) preorder(child); }
Whatever “visit” means

47. a c b R X n d p Z i g f h j i e c b R d j g f h Preorder Traversal
General Trees
Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
a c b R X n d p Z i g f h j
Visit the parent then visit its children
preorder(Node<E> node)
if (node != null) {
visit(node);
for (Node<E> child : node.children()) preorder(child); }
Whatever “visit” means

48. O(n) a c b R X n d p Z i g f h j i e c b R d j g f h
Preorder Traversal
General Trees
Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
a c b R X n d p Z i g f h j
Visit the parent then visit its children
preorder(Node<E> node)
if (node != null) {
visit(node);
for (Node<E> child : node.children()) preorder(child); }
O(n)
Whatever “visit” means

49. ADS2 Lecture 14

50. i e c b R d j g f h Parenthetic Representation General Trees
Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
(a (c (b ((R),(X),(n))),(d)),(p),(Z),(i (g ((f),(h))),(j)))
Also called Caley Notation (I think) and is a preorder print

51. i e c b R d j g f h Postorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the children then visit the parent
Typical use is we want to “assemble” parts
An actual use: du in unix

52. i e c b R d j g f h Postorder Traversal General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the children then visit the parent
postorder(Node<E> node)
if (node != null) {
for (Node<E> child : node.children()) postorder(child);
visit(node); }

53. O(n) i e c b R d j g f h Postorder Traversal General Trees
Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit the children then visit the parent
prostorder(Node<E> node)
if (node != null) {
for (Node<E> child : node.children()) postorder(child);
visit(node); }
O(n)

54. O(n) R X n b d c p Z f h g j i a i e c b R d j g f h
Postorder Traversal
General Trees
Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
R X n b d c p Z f h g j i a
Visit the children then visit the parent
prostorder(Node<E> node)
if (node != null) {
for (Node<E> child : node.children()) postorder(child);
visit(node); }
O(n)

55. ADS2 Lecture 14

56. i e c b R d j g f h Other Traversals General Trees Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
Visit depth 0 nodes
Visit depth 1 nodes …
Visit depth height nodes
Also Known As (aka)
Breadth First Search (bfs)

57. O(n) a c p Z i b d g j R X n f h i e c b R d j g f h Other Traversals
General Trees
Tree<E>
Node<E> root i e c b R d j g f h a X n p Z
Node:
<E> element
Node<E> parent
ArrayList<Node<E>> children
a c p Z i b d g j R X n f h
Visit depth 0 nodes
Visit depth 1 nodes …
Visit depth height nodes
O(n)
Also Known As (aka)
Breadth First Search (bfs)
Note: preorder = dfs!

58. Binary Trees
ADS2 Lecture 14

59. Binary Trees A binary tree is a tree in which each node has
A reference to a left node
A value
A reference to a right node
A binary tree is either empty or
Contains a single node r (the root)
whose left and right subtrees are binary trees
The tree is accessed via a reference to the root.
The nodes with both child references null are the leaves.
recursive definition!
ADS2 Lecture 14

60. nodes
intenal nodes 0
leaf nodes
edges
height ?

61. nodes
intenal nodes 0
leaf nodes
edges
height

62. nodes
intenal nodes 1
leaf nodes
edges
height

63. nodes
intenal nodes 3
leaf nodes
edges
height

64. nodes
intenal nodes 7
leaf nodes
edges
height

65. nodes n
intenal nodes (nI) h ≤ nI ≤ 2h-1
leaf nodes (nE) ≤ nE ≤ 2h
edges (e) e = n-1
height (h) h+1 ≤ n ≤2h+1-1
height (h) log2(n+1)-1 ≤ h ≤ n-1
nodes
intenal nodes 7
leaf nodes
edges
height

66. And this is crucial
height (h) log2(n+1)-1 ≤ h ≤ n-1

67. A Forest! t m w p u z v

68. This is also a binary tree3 5 7 9 11

69. Balance A typical binary tree: This tree is quite well balanced.
An extreme unbalanced tree might have no right pointers
- would look more like a linked list.
Generally tree-based algorithms work most efficiently
on balanced trees.
A binary tree in which no value occurs in more than one node is injective. We deal only with injective binary trees.
ADS2 Lecture 14

70. Node of binary tree Binary tree node: A question:
How would we represent the nodes of a tree (not binary)?
We don’t know how many children each node has
Binary tree node:
public class BTNode<E>{
private E element;
private BTNode<E> left;
private BTNode<E> right;
/**Creates a node with null references*/
public BTNode(){
this(null,null,null); }
/** Creates node with the given element, L and R nodes*/
public BTNode(E e, BTNode<E> l,BTNode<E> r){
element = e;
left=l;
right=r;
plus getters and setters
ADS2 Lecture 14

71. For a binary tree of strings

72. For a binary tree of strings

73. For a binary tree of strings

74. For a binary tree of strings

75. For a binary tree of strings
And what is that?

76. For a binary tree of strings

77. Definitions p - a node of a binary tree
p.left node of a binary tree - the left subtree
p.right node of a binary tree - the right subtree
if p.left = q - p is the parent of q and q is the left child of p
if p.right = q - p is the parent of q and q is the right child of p. p
Left subtree of p
Right subtree of p
ADS2 Lecture 14

78. Traversals
A traversal of a binary tree is a sequence of nodes of the tree.
Special traversals:
1. Inorder traversal - defined recursively
The inorder traversal of an empty tree is the empty sequence
The inorder traversal of a non-empty tree is:
the inorder traversal of the left subtree (a sequence)
the root value (a sequence with one member)
the inorder traversal of the right subtree (a sequence)
ADS2 Lecture 14

79. Example (inorder traversal)
Inorder traversal is a b c d e f g h i j
ADS2 Lecture 14

80. Traversals contd. 2. Preorder traversal – defined recursively
The preorder traversal of an empty tree is the empty sequence
The preorder traversal of a non-empty tree is:
the root value (a sequence with one member)
the preorder traversal of the left subtree (a sequence)
the preorder traversal of the right subtree (a sequence)
Preorder traversal is e c b a d i g f h j
ADS2 Lecture 14

81. Traversals contd. Postorder traversal defined recursively:
The postorder traversal of an empty tree is the empty sequence
The postorder traversal of a non-empty tree is:
the postorder traversal of the left subtree (a sequence)
the postorder traversal of the right subtree (a sequence)
the root value (a sequence with one member)
Postorder traversal is a b d c f h g j i e
ADS2 Lecture 14

84. An Euler Walk

85. An ArrayList Representation of a Binary Tree
Nodes have an integer position
We put the nodes in an Array or ArrayList
ADS2 Lecture 14

86. An ArrayList Representation of a Binary Tree3 1 2 4 8 5 7 6 12 13
ADS2 Lecture 14

87. An ArrayList Representation of a Binary Tree3 1 2 4 8 5 7 6 12 13
We have an ArrayList<Node<E>> T
A Node<E> has an integer position attribute
The root has position 1, i.e. T[1] is the root node
T[i].position() == i
ADS2 Lecture 14

88. An ArrayList Representation of a Binary Tree3 1 2 4 8 5 7 6 12 13
We have an ArrayList<Node<E>> T
A Node<E> has an integer position attribute
The root has position 1, i.e. T[1] is the root node
T[i].position() == i
Left of T[i] is T[i*2]
Right of T[i] is T[i*2 + 1]
Parent of T[i] is T[i/2]
ADS2 Lecture 14

89. Putting an expression into a tree An Expression Tree (or species tree)
ADS2 Lecture 14

90. Putting an expression into a tree An Expression Tree (or species tree)
We use two stacks
S1 of Node<E>
S2 of String
ADS2 Lecture 14

91. Putting an expression into a tree An Expression Tree (or species tree)
We use two stacks
S1 of Node<E>
S2 of String
if we read “(“ S1.push(new Node(null,”!”,null))
if we read is in {+,-,*,/} S2.push(operator)
if we read “)”
Node r = S1.pop()
Node l = S1.pop()
Node node = S1.pop()
String op = S1.pop()
node.setLeft(l); node.setElement(op); node.setRight(r)
S1.push(node)
if we read String s and it is something else (a leaf) S1.push(new Node(null,s,null))
ADS2 Lecture 14

92. No Expense Spared
ADS2 Lecture 14

93. .