The document provides information about different types of trees: - A tree is a connected undirected graph without cycles. It has a unique path between any two vertices. - Rooted trees have one distinguished root vertex. They define the level, height and relationships between parent/child and ancestor/descendant vertices. - Binary trees restrict each vertex to having zero, one or two children. Ordered binary trees further define the left and right children. - Expression trees represent mathematical expressions, with operators in internal nodes and operands in leaves. Their structure dictates evaluation order. - Other tree properties discussed include those of full/complete binary trees and algorithms for finding minimum spanning trees like Prim's and K

1. Discrete Mathematics
NOR AIDAWATI BINTI ABDILLAH
TREES
1

2. Tree
Definition 1. A tree is a connected
undirected graph with no simple circuits.
Theorem 1. An undirected graph is a tree if
and only if there is a unique simple path
between any two of its vertices.
2

3. Which graphs are trees?
a) b)
c)
3

4. Is it a tree?
NO!
yes! All the nodes are yes! (but not
not connected a binary tree)
NO!
There is a cycle yes! (it’s actually
and an extra the same graph as
edge (5 nodes the blue one) – but
and 5 edges) usually we draw
tree by its “levels”4

5. Rooted Trees
q Rooted tree is a tree in which one vertex is
distinguished and called a root
q Level of a vertex is the number of edges
between the vertex and the root
q The height of a rooted tree is the
maximum level of any vertex
q Children, siblings and parent vertices in a
rooted tree
q Ancestor, descendant relationship
between vertices
5

6. q The parent of a non-root vertex is the unique vertex
u with a directed edge from u to v.
q A vertex is called a leaf if it has no children.
q The ancestors of a non-root vertex are all the
vertices in the path from root to this vertex.
q The descendants of vertex v are all the vertices that
have v as an ancestor.
q The level of vertex v in a rooted tree is the length of
the unique path from the root to v.
q The height of a rooted tree is the maximum of the
6
levels of its vertices.

7. A Tree Has a Root
root node
a
internal vertex parent of g
b c
d e f g
leaf
siblings
h i
7

8. a
b c
d e f g
ancestors of h and i
h i

9. The level of a vertex v in a rooted tree is
the length of the unique path from the root
to this vertex.
level 2
level 3

10. a
b c
d e f g
subtree with b as its
h i root
subtree with c as its
root

11. Tree Properties
q There is one and only one path between
every pair of vertices in a tree, T.
q A tree with n vertices has n-1 edges.
q Any connected graph with n vertices and n-1
edges is a tree.
q A graph is a tree if and only if it is minimally
connected. 11

12. Tree Properties
Theorem . There are at most 2 H leaves in a
binary tree of height H.
Corallary. If a binary tree with L leaves is
full and balanced, then its height is
H =  log2 L  .
12

13. Properties of Trees
There are at most mh leaves in an m-ary
tree of height h.
A rooted m-ary tree of height h is called
balanced if all leaves are at levels h or h-1.

14. Properties of Trees
A full m-ary tree with
(i) n vertices has i = (n-1)/m internal
vertices and l = [(m-1)n+1]/m leaves.
(ii) i internal vertices has n = mi + 1
vertices and l = (m-1)i + 1 leaves.
(iii) l leaves has n = (ml - 1)/(m-1) vertices
and i = (l-1)/(m-1) internal vertices.

15. Properties of Trees
A full m-ary tree with i internal vertices
contains n = mi+1 vertices.

16. Proof
q We know n = mi+1 (previous
theorem) and n = l+i,
q n – no. vertices
q i – no. internal vertices
q l – no. leaves
q For example, i = (n-1)/m

17. Binary Tree
A rooted tree in which each vertex has either
no children, one child or two children.
The tree is called a full binary tree if every
internal vertex has exactly 2 children.
A
B C right child of A
left subtree
of A D E F G
right
subtree of
H C 17
I J

18. Ordered Binary Tree
Definition 2’’. An ordered rooted tree is a
rooted tree where the children of each
internal vertex are ordered.
In an ordered binary tree, the two
possible children of a vertex are called
the left child and the right child, if they
exist.
18

19. An Ordered Binary Tree
A
B
E C
K F G D
L H
M I
J
19

20. Is this binary tree balanced?
A rooted binary tree of height
Lou H is called balanced if all its
leaves are at levels H or H-1.
Hal Max
Ed Ken Sue
Joe Ted
20

21. Searching takes time . . .
So the goal in computer programs is to find
any stored item efficiently when all stored
items are ordered.
A Binary Search Tree can be used to store
items in its vertices. It enables efficient
searches.
21

22. A Binary Search Tree (BST) is . . .
A special kind of binary tree in which:
1. Each vertex contains a distinct key value,
2. The key values in the tree can be compared using
“greater than” and “less than”, and
3. The key value of each vertex in the tree is
less than every key value in its right subtree, and
greater than every key value in its left subtree.

23. A Binary Expression Tree is . . .
A special kind of binary tree in which:
l Each leaf node contains a single operand,
l Each nonleaf node contains a single binary
operator, and
l The left and right subtrees of an operator
node represent subexpressions that must be
evaluated before applying the operator at
the root of the subtree.
23

24. Expression Tree
q Each node contains an
operator or an operand
q Operands are stored in
leaf nodes
q Parentheses are not stored (x + y)*((a + b)/c)
in the tree because the tree structure
dictates the order of operand evaluation
q Operators in nodes at higher levels are
evaluated after operators in nodes at lower
levels
24

25. A Binary Expression Tree
‘*’
‘+’ ‘3’
‘4’ ‘2’
What value does it have?
( 4 + 2 ) * 3 = 18
25

26. A Binary Expression Tree
‘*’
‘+’ ‘3’
‘4’ ‘2’
Infix: ((4+2)*3)
Prefix: * + 4 2 3 evaluate from right
Postfix: 4 2 + 3 * evaluate from left
26

27. Levels Indicate Precedence
When a binary expression tree is used to
represent an expression, the levels of the
nodes in the tree indicate their relative
precedence of evaluation.
Operations at higher levels of the tree are
evaluated later than those below them.
The operation at the root is always the
last operation performed.
27

28. Evaluate
this binary expression tree
‘*’
‘-’ ‘/’
‘8’ ‘5’ ‘+’ ‘3’
‘4’ ‘2’
What expressions does it represent?
28

29. A binary expression tree
‘*’
‘-’ ‘/’
‘8’ ‘5’ ‘+’ ‘3’
‘4’ ‘2’
Infix: ((8-5)*((4+2)/3))
Prefix: *-85 /+423 evaluate from right
Postfix: 85- 42+3/* evaluate from left
29

30. Binary Tree for Expressions
30

31. Complete Binary Tree
Also be defined as a full binary tree in which all
leaves are at depth n or n-1 for some n.
In order for a tree to be the latter kind of complete
binary tree, all the children on the last level must
occupy the leftmost spots consecutively, with no
spot left unoccupied in between any two
A A
B B C
C
D F G D E F G
E
H I J K L M N O
H I 31
Complete binary tree Full binary tree of depth 4

32. Difference between binary and
complete binary tree
BINARY TREE ISN'T NECESSARY THAT ALL OF
LEAF NODE IN SAME LEVEL BUT
COMPLETE BINARY TREE MUST HAVE ALL LEAF
NODE IN SAME LEVEL.
Binary Tree
32
Complete Binary Tree

33. SPANNING TREES
q A spanning tree of a connected graph G is
a sub graph that is a tree and that includes
every vertex of G.
q A minimum spanning tree of a weighted
graph is a spanning tree of least weight
(the sum of the weights of all its edges is
least among all spanning tree).
q Think: “smallest set of edges needed to
connect everything together”
33

34. A graph G and three of its
spanning tree
We can delete any edge without deleting any vertex (to remove
the cycle), but leave the graph connected.
34

35. PRIM’S ALGORITHM
q Choose any edge with smallest weight,
putting it into the spanning tree.
q Successively add to the tree edges of
minimum weight that are incident to a
vertex already in the tree and not forming a
simple circuit with those edges already in
the tree.
q Stop when n – 1 edges have been added.
35

36. EXAMPLE
PRIM’S ALGORITHM
Use Prim’s algorithm to find a minimum spanning tree
in the weighted graph below:
a 2 b 3 c 1 d
3 1 2 5
e 4 f 3 g 3 h
4 2 4 3
i 3 j 3 k 1 l 36

37. SOLUTION
Choice Edge Weight
1 {b, f} 1
a 2 b c 1 d 2 {a, b} 2
3 {f, j} 2
3 1 2 4 {a, e} 3
f 3 g 3 h 5 {i, j} 3
e
6 {f, g} 3
2 3 7 {c, g} 2
8 {c, d} 1
i j 9 {g, h} 3
3 k 1 l
10 {h, l} 3
11 {k, l} 1
Total: 24
37

38. KRUSKAL’S ALGORITHM
q Choose an edge in the graph with
minimum weight.
q Successively add edges with minimum
weight that do not form a simple circuit
with those edges already chosen.
q Stop after n – 1 edges have been
selected.
38

39. Kruskal’s Algorithm
q Pick the cheapest link (edge)
available and mark it
q Pick the next cheapest link available
and mark it again
q Continue picking and marking link
that does not create the circuit
***Kruskal’s algorithm is efficient and
optimal 39

40. EXAMPLE
KRUSKAL’S ALGORITHM
Use Kruskal’s algorithm to find a minimum spanning
tree in the weighted graph below:
a 2 b 3 c 1 d
3 1 2 5
e 4 f 3 g 3 h
4 2 4 3
i 3 j 3 k 1 l 40

41. SOLUTION
Choice Edge Weight
1 {c, d} 1
2 {k, l} 1
a 2 b 3 c 1 d 3 {b, f} 1
4 {c, g} 2
3 1 2 5 {a, b} 2
f g 3 h 6 {f, j} 2
e
7 {b, c} 3
2 8 {j, k} 3
9 {g, h} 3
i j 3 10 {i, j} 3
3 k 1 l
11 {a, e} 3
Total: 24
41

42. TRAVELLING SALESMAN
PROBLEM (TSP)
The goal of the Traveling Salesman Problem
(TSP) is to find the “cheapest” tour of a select
number of “cities” with the following
restrictions:
●
You must visit each city once and only once
●
You must return to the original starting point
42

43. TSP
TSP is similar to these variations of Hamiltonian Circuit
problems:
●
Find the shortest Hamiltonian cycle in a
weighted graph.
●
Find the Hamiltonian cycle in a weighted
graph with the minimal length of the
longest edge. (bottleneck TSP).
A route returning to the beginning is known as a
Hamiltonian Circuit
A route not returning to the beginning is known as a
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Hamiltonian Path

44. THANK YOU
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