Trees are useful for representing hierarchically ordered data like directory structures and family trees. A tree is a collection of nodes connected by branches, with each node having at most one parent. It is defined recursively as a root node with zero or more subtrees. Binary trees restrict nodes to having at most two children. Trees can be traversed in preorder, inorder, or postorder. Binary search trees organize nodes to keep all left descendants less than the parent and all right descendants greater than the parent. Graphs generalize trees by allowing many-to-many relationships between nodes.

1. Introduction to Trees

2. Linear Lists and Trees
• Linear lists (arrays, linked list)are useful for serially ordered
data
– (e1,e2,e3,…,en)
– Days of week
– Months in a year
– Students in a class
• Trees are useful for hierarchically ordered data
– directory structure
– Parenthetical Listing
– family trees:
• all descendants of a particular person
• all ancestors born after year 1800 of a particular person

3. Parenthetical Listing
A (B (C D) E F (G H I) )

4.  Tree
 A tree is a collection of Nodes
 and a finite set of directed lines called branches, that connect the nodes.
 Each node can have 0 or more children
 A node can have at most one parent
 Tree defined recursively
 A tree consists of a distinguished node r, called the root, and zero or more
non-empty (sub) trees T1, T2, …, Tk each of whose roots are connected by
a directed edge from r.
 A tree is a collection of N nodes, one of which is the root and N-1 edges.
 The number of branches associated with a node is the degree of the
node.
 Binary tree
 Tree with 0–2 children per node

5.  Terminology
 Root  no parent
 Leaf  no child
 Interior  non-leaf
 Height  distance from root to leaf
 The root of each sub tree is said to be a child of r and r is
said to be the parent of each sub tree root.
 Siblings  nodes with the same parent

6. • Depth (of node): the length of the unique path from the root to a
node.
• Depth (of tree): The depth of a tree is equal to the depth of its
deepest leaf.
• Height (of node): the length of the longest path from a node to a
leaf.
– All leaves have a height of 0
– The height of the root is equal to the depth of the tree
• When the branch is directed toward the node, it is indegree branch.
• When the branch is directed away from the node, it is an outdegree
branch.
• The sum of the indegree and outdegree branches is the degree of the
node.

9. Recursive definition of a tree
• A tree is a set of nodes that either:
- is empty or
- has a designated node, called the root, from
which hierarchically descend zero or more sub
trees, which are also trees.
9

10. Binary trees
• A binary tree is a tree in which no node can
have more than two subtrees; the maximum
outdegree for a node is two.
• In other words, a node can have zero, one, or
two subtrees.
• These subtrees are designated as the left
subtree and the right subtree.

14. Picture of a binary tree
a
b c
d e
g h i
l
f
j k

15. Tree traversals

16. Preorder traversal
• In preorder, the root is visited first
• Here’s a preorder traversal to print out all
the elements in the binary tree:
A
B C
D E F G
A B D E C F G

17. Example1

19. Example2

21. Inorder traversal
• In inorder, the root is visited in the middle
• Here’s an inorder traversal to print out all
the elements in the binary tree:

22. Example1

23. Example1

24. Example2

26. Postorder traversal
• In Postorder, the root is visited last
• Here’s a Postorder traversal to print out all
the elements in the binary tree:

27. Example1

28. Example1

29. Example2

31. Arithmetic Expression Tree
• Binary tree for an arithmetic expression
– internal nodes: operators
– leaves: operands
• Example: arithmetic expression tree for the expression
((2  (5 - 1)) + (3  2))
+

-2
5 1
3 2

32. Write down prefix, Infix & Postfix expression

33. Write down prefix, Infix & Postfix expression

34. Binary Search Trees
• Binary Search Tree Property:
– the value stored at the root of a sub tree is greater than any value in its
left sub tree and less than any value in its right sub tree!
• Where is the smallest element?
– leftmost element
• Where is the largest element?
– rightmost element
• How to build & maintain binary trees?
– Insertion
– Deletion
• Maintain key property (invariant)
– Smaller values in left sub tree
– Larger values in right sub tree

35. Binary Search Trees
• Key property
– Value at node
• Smaller values in left subtree
• Larger values in right subtree
– Example
• X > Y
• X < Z
Y
X
Z

36. Binary Search Trees
• Examples
Binary search
trees
Not a binary
search tree
5
10
30
2 25 45
5
10
45
2 25 30
5
10
30
2
25
45

37. Binary Search Tree – Insertion
• Algorithm
1. Perform search for value X
2. Search will end at node Y (if X not in tree)
3. If X < Y, insert new leaf X as new left sub tree for Y
4. If X > Y, insert new leaf X as new right sub tree for Y
• Observations
– O( log(n) ) operation for balanced tree

38. Example Insertion
• Insert ( 20 )

39. Binary Search Tree – Deletion
Algorithm
1. Perform search for value X
2. If X is a leaf, delete X
3. Else // must delete internal node
a) Replace with largest value Y on left sub tree
OR smallest value Z on right sub tree
b) Delete replacement value (Y or Z) from sub tree
Observation
- O( log(n) ) operation for balanced tree

40. Example Deletion (Leaf)
• Delete ( 25 )
5
10
30
2 25 45
10 < 25, right
30 > 25, left
25 = 25, delete
5
10
30
2 45

41. Example Deletion (Internal Node)
• Delete ( 10 )
5
10
30
2 25 45
5
5
30
2 25 45
2
5
30
2 25 45
Replacing 10
with largest
value in left sub
tree
Replacing 5 with
largest value in
left sub tree
Deleting leaf

42. Example Deletion (Internal Node)
• Delete ( 10 )
5
10
30
2 25 45
5
25
30
2 25 45
5
25
30
2 45
Replacing 10
with smallest
value in right
sub tree
Deleting leaf Resulting tree

43. Iterative Search of Binary Tree
Node *Find( Node *n, int key) {
while (n != NULL) {
if (n->data == key) // Found it
return n;
if (n->data > key) // In left subtree
n = n->left;
else // In right subtree
n = n->right;
}
return null;
}
Node * n = Find( root, 5);

44. Recursive Search of Binary Tree
Node *Find( Node *n, int key) {
if (n == NULL) // Not found
return( n );
else if (n->data == key) // Found it
return( n );
else if (n->data > key) // In left subtree
return Find( n->left, key );
else // In right subtree
return Find( n->right, key );
}
Node * n = Find( root, 5);

45. Example Binary Searches
• Find ( root, 2 )
5
10
30
2 25 45
5
10
30
2
25
45
10 > 2, left
5 > 2, left
2 = 2, found
5 > 2, left
2 = 2, found
root

46. Example Binary Searches
• Find (root, 25 )
5
10
30
2 25 45
5
10
30
2
25
45
10 < 25, right
30 > 25, left
25 = 25, found
5 < 25, right
45 > 25, left
30 > 25, left
10 < 25, right
25 = 25, found

47. Types of Binary Trees
• Degenerate – only one child
• Balanced – “mostly” two children
• Complete – always two children
Degenerate
binary tree
Balanced
binary tree
Complete
binary tree

48. Introduction to Graph
• What is a graph?
– A data structure that consists of a set of nodes
(vertices) and a set of edges that relate the nodes to
each other
• The set of edges describes relationships among
the vertices
• A graph G is defined as follows:
G=(V,E)
V(G): a finite, nonempty set of vertices
E(G): a set of edges (pairs of vertices)

49. Directed vs. undirected graphs
• When the edges in a graph have no direction,
the graph is called undirected

50. Directed vs. undirected graphs (cont.)
• When the edges in a graph have a direction,
the graph is called directed (or digraph)
Warning: if the graph is directed, the order of the vertices in each edge is
important !!

51. Trees vs graphs
Trees are special cases of graphs!!

52. Graph terminology
• Adjacent nodes: two nodes are adjacent if they are connected
by an edge
– 5 is adjacent to 7
– 7 is adjacent from 5
• Path: a sequence of vertices that connect two nodes in a
graph
• Complete graph: a graph in which every
vertex is directly connected to
every other vertex

53. Graph terminology
• What is the number of edges in a complete
directed graph with N vertices?
N * (N-1)

54. Graph terminology
• What is the number of edges in a complete
undirected graph with N vertices?
N * (N-1) / 2

55. Graph terminology
• Weighted graph: a graph in which each edge
carries a value

56. Graph implementation
• Array-based implementation
– A 1D array is used to represent the vertices
– A 2D array (adjacency matrix) is used to represent
the edges