Trees (slides)

Definition: Let 𝐴𝐴 be a set, and let 𝑇𝑇 be a relation on 𝐴𝐴.
We say that 𝑇𝑇 is a tree if there is a vertex 𝑣𝑣0 in 𝐴𝐴 with
the property that
1) there exists a unique path in 𝑇𝑇 from 𝑣𝑣0 to every
other vertex in 𝐴𝐴,
2) no path from 𝑣𝑣0 to 𝑣𝑣0.
Definition: 𝑣𝑣0 is called the root of the tree, and T is
referred to as a rooted tree denoted by 𝑇𝑇, 𝑣𝑣0 .
Trees
© S. Turaev, CSC 1700 Discrete Mathematics 2

Theorem: Let 𝑇𝑇, 𝑣𝑣0 be a rooted tree. Then
1. There are no cycle in 𝑇𝑇.
2. Vertex 𝑣𝑣0 is the only root of 𝑇𝑇.
3. Each vertex in 𝑇𝑇, other than 𝑣𝑣0,
has in-degree one, and 𝑣𝑣0 has in-degree zero.
Trees

Levels, Parent-Offspring, Siblings
© S. Turaev, CSC 1700 Discrete Mathematics
2
1
5
3 4
6
7
Level 1 vertices: the vertices of
the edges beginning at 𝑣𝑣0 (level 0)
Level 𝑘𝑘 vertices: the vertices of the
edges beginning at those level 𝑘𝑘 − 1
vertices
Parent-offspring: for all pairs (𝑎𝑎, 𝑏𝑏) in 𝑇𝑇,
𝑎𝑎 is called the parent of 𝑏𝑏 and 𝑏𝑏 is called
the offspring of 𝑎𝑎
Siblings: the vertices that have the same
parent
Height of a tree: the largest level number
of a tree
Leaves: the vertices that have no
offspring
4

Levels, Parent-Offspring, Siblings
2
1
5
3 4
6
7
Level 0
Level 1
Level 2
Level 3
Height 3Leaf
Root
Child /
Offspring
Parent
Siblings
5

Theorem: Let 𝑇𝑇, 𝑣𝑣0 be a rooted tree. Then
 𝑇𝑇 is irreflexive
 𝑇𝑇 is asymmetric
 If 𝑎𝑎, 𝑏𝑏 in 𝑇𝑇 and 𝑏𝑏, 𝑐𝑐 in 𝑇𝑇, then 𝑎𝑎, 𝑐𝑐 is not
in 𝑇𝑇, for all 𝑎𝑎, 𝑏𝑏 and 𝑐𝑐 in 𝐴𝐴.
Trees

Example: Let 𝐴𝐴 = 𝑣𝑣1, 𝑣𝑣2, 𝑣𝑣3, … , 𝑣𝑣10 and let
𝑇𝑇 =
𝑣𝑣2, 𝑣𝑣3 , 𝑣𝑣2, 𝑣𝑣1 , 𝑣𝑣4, 𝑣𝑣5 , 𝑣𝑣4, 𝑣𝑣6 ,
𝑣𝑣5, 𝑣𝑣8 , 𝑣𝑣6, 𝑣𝑣7 , 𝑣𝑣4, 𝑣𝑣2 , 𝑣𝑣7, 𝑣𝑣9 , 𝑣𝑣7, 𝑣𝑣10
Show that 𝑇𝑇 is a rooted tree and identify the root.
Trees

Definition:
 If 𝑛𝑛 is a positive integer, we say that a tree is an 𝑛𝑛-
tree if every vertex has at most 𝑛𝑛 offspring.
 A 2-tree is called a binary tree.
Definition:
 If all vertices of 𝑇𝑇, other than the leaves, have
exactly 𝑛𝑛 offspring, we say that 𝑇𝑇 is a complete
𝑛𝑛-tree.
 A complete 2-tree is called a completed binary tree.
𝒏𝒏-trees

 Let 𝑇𝑇, 𝑣𝑣0 be a rooted tree on the set 𝐴𝐴, and let 𝑣𝑣
be a vertex of 𝑇𝑇.
 Let 𝐵𝐵 be the set consisting of 𝑣𝑣 and all its
descendants, i.e., all vertices of 𝑇𝑇 that can be reached
by a path beginning at 𝑣𝑣.
 Let 𝑇𝑇 𝑣𝑣 be the restriction of the relation 𝑇𝑇 to 𝐵𝐵,
that is 𝑇𝑇 ∩ (𝐵𝐵 × 𝐵𝐵).
 Delete all vertices that are not descendants of 𝑣𝑣 and all
edges that do not begin and end at any such vertex.
Subtrees

Theorem: If 𝑇𝑇, 𝑣𝑣0 is a rooted tree and 𝑣𝑣 in 𝑇𝑇, then
𝑇𝑇 𝑣𝑣 is also a rooted tree with root 𝑣𝑣.
We will say that 𝑇𝑇 𝑣𝑣 is the subtree of 𝑇𝑇 beginning at 𝑣𝑣.
Subtrees
2 31
0
4 5
6
10

Theorem: If 𝑇𝑇, 𝑣𝑣0 is a rooted tree and 𝑣𝑣 in 𝑇𝑇, then
𝑇𝑇 𝑣𝑣 is also a rooted tree with root 𝑣𝑣.
We will say that 𝑇𝑇 𝑣𝑣 is the subtree of 𝑇𝑇 beginning at 𝑣𝑣.
Subtrees
2
0
4 5
6
11

Exercise 1: Determine if 𝑅𝑅 is a tree and, if it is, find the
root.
 𝐴𝐴 = 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑, 𝑒𝑒
𝑅𝑅 = 𝑎𝑎, 𝑑𝑑 , 𝑏𝑏, 𝑐𝑐 , 𝑐𝑐, 𝑎𝑎 , 𝑑𝑑, 𝑒𝑒
 𝐴𝐴 = 1, 2, 3, 4, 5, 6
𝑅𝑅 = 2,1 , 3,4 , 5,2 , 6,5 , 6,3
Exercises

Exercise 2: Consider the rooted tree 𝑇𝑇, 0 .
Exercises
0
1 2
4 5 6 8 9
10
11 12
7
3
13 14
15
13

Exercise 2: Consider the rooted tree 𝑇𝑇, 0 .
1. List all level-3 vertices
2. List all leaves
3. What are the siblings of 8?
4. What are the descendants of 3?
5. Compute 𝑇𝑇 2
6. Compute 𝑇𝑇 3
7. What is the height of 𝑇𝑇, 0 ?
Exercises

Example: Use a tree to denote the following algebraic
expression
3 − 2 × 𝑏𝑏 + 𝑏𝑏 − 2 − 3 + 𝑏𝑏
Labeled Trees
+
- -
3
2 b
x -
b 2 3 b
+
15

Example: Use a tree to denote the following algebraic
expression
3 × 1 − 𝑎𝑎 ÷ 4 + 7 − 𝑏𝑏 + 2 × 7 + 𝑎𝑎 ÷ 𝑏𝑏
Labeled Trees

Positional 𝑛𝑛-tree:
• 𝑛𝑛-tree: every vertex has at most 𝑛𝑛 offspring
• positional 𝑛𝑛-tree: label the offspring of a given
vertex from left to right with numbers 1,2, … , 𝑛𝑛
• some of the offspring in the sequence may be
missing
Labeled Trees

Example: positional 3-tree:
Labeled Trees
2
2
3
3
3
1 1
3
2 31
2 31
18

Example: positional 2-tree:
Labeled Trees
L R
L
R
R L
L L R
R
19

Visiting
Performing appropriate tasks at a vertex will be
called visiting the vertex.
Tree search
The process of visiting each vertex of a tree in some
specific order will be called searching the tree or
performing a tree search.
Tree Searching

Algorithm PREORDER
Step 1: Visit 𝑣𝑣
Step 2: If 𝑣𝑣𝐿𝐿 exists, then apply this algorithm to
𝑇𝑇 𝑣𝑣𝐿𝐿 , 𝑣𝑣𝐿𝐿
Step 3: If 𝑣𝑣𝑅𝑅 exists, then apply this algorithm to
𝑇𝑇 𝑣𝑣𝑅𝑅 , 𝑣𝑣𝑅𝑅
Tree Searching

Example 1
Tree Searching
A
B
C
D F
E
G J
I
L
K
H
1
2
3
4
5 6
7
8
9
10
11
A B C D E F G H I J K L
22

Example 2: 𝑎𝑎 − 𝑏𝑏 × 𝑐𝑐 + 𝑑𝑑/𝑒𝑒
Tree Searching
×
-
a b
e
/
+
c
d
× - a b + c / d e
1
2 3
4
5
7 8
6
23

Prefix or Polish form:
× − 𝑎𝑎 𝑏𝑏 + 𝑐𝑐 / 𝑑𝑑 𝑒𝑒 (𝑎𝑎 = 6, 𝑏𝑏 = 4, 𝑐𝑐 = 5, 𝑑𝑑 = 2, 𝑒𝑒 = 2)
1. × −6 4 + 5 / 2 2
2. × 2 + 5 / 2 2 replacing −6 4 by 2 since 6 − 4 = 2
3. × 2 + 5 1 replacing / 2 2 by 1 since 2/2 = 1
4. × 2 6 replacing + 5 1 by 6 since 5 + 1 = 6
5. 12 replacing × 2 6 by 12 since
2 × 6 = 12
Tree Searching

Algorithm INORDER
Step 1: Search the left subtree 𝑇𝑇 𝑣𝑣𝐿𝐿 , 𝑣𝑣𝐿𝐿 , if it exists
Step 2: Visit the root 𝑣𝑣
Step 3: Search the right subtree 𝑇𝑇 𝑣𝑣𝑅𝑅 , 𝑣𝑣𝑅𝑅 , if it exists
Tree Searching

Algorithm POSTORDER
Step 1: Search the left subtree 𝑇𝑇 𝑣𝑣𝐿𝐿 , 𝑣𝑣𝐿𝐿 , if it exists
Step 2: Search the right subtree 𝑇𝑇 𝑣𝑣𝑅𝑅 , 𝑣𝑣𝑅𝑅 , if it exists
Step 3: Visit the root 𝑣𝑣
Tree Searching

Example: Traveling the tree using INORDER and
POSTORDER
𝑎𝑎 − 𝑏𝑏 × 𝑐𝑐 + 𝑑𝑑/𝑒𝑒
Tree Searching
×
-
a b
e
/
+
c
d
INORDER: 𝑎𝑎 − 𝑏𝑏 × 𝑐𝑐 + 𝑑𝑑/𝑒𝑒
POSTORDER: 𝑎𝑎 𝑏𝑏 − 𝑐𝑐 𝑑𝑑 𝑒𝑒 / +×
27

Infix notation: Algebraic symbols lie between their
arguments
𝑎𝑎 − 𝑏𝑏 × 𝑐𝑐 + 𝑑𝑑 / 𝑒𝑒
(𝑎𝑎 − 𝑏𝑏) × (𝑐𝑐 + (𝑑𝑑/𝑒𝑒))
or
𝑎𝑎 − (𝑏𝑏 × ( 𝑐𝑐 + 𝑑𝑑 /𝑒𝑒))
Tree Searching

Postfix or reverse Polish:
𝑎𝑎 𝑏𝑏 − 𝑐𝑐 𝑑𝑑 𝑒𝑒 / + × (𝑎𝑎 = 2, 𝑏𝑏 = 1, 𝑐𝑐 = 3, 𝑑𝑑 = 4, 𝑒𝑒 = 2)
1. 2 1 − 3 4 2 / + ×
2. 1 3 4 2 / + × replacing 2 1 − with 1 since 2 − 1 = 1
3. 1 3 2 + × replacing 4 2 / with 2 since 4/2 = 2
4. 1 5 × replacing 3 2 + with 5 since 3 + 2 = 5
5. 5 replacing 1 5 × with 5 since 1 × 5 =5
Tree Searching

Show the result of performing a preorder search of the
tree
Exercises
x
y
s
z
t
u
v
30

Show the result of performing an inorder search of the
tree
Exercises
x
y
s
z
t
u
v
31

Show the result of performing a postorder search of the
tree
Exercises
x
y
s
z
t
u
v
32

Show the result of performing preorder, inorder and
postorder searches of the tree
𝑇𝑇 =
𝑎𝑎, 𝑏𝑏 , 𝑎𝑎, 𝑑𝑑 , 𝑏𝑏, 𝑐𝑐 , 𝑏𝑏, 𝑖𝑖 , 𝑑𝑑, 𝑘𝑘 , 𝑑𝑑, 𝑒𝑒 ,
𝑐𝑐, 𝑔𝑔 , 𝑐𝑐, ℎ , 𝑒𝑒, 𝑗𝑗 , 𝑒𝑒, 𝑓𝑓
Exercises

Show the result of performing preorder, inorder and
postorder searches of the tree
𝑇𝑇 = 1,2 , 1,3 , 2,4 , 3,5 , 4,6 , 5,7
Exercises

Evaluate the expressions, which are given in Polish, or
prefix, notation
• × − + 3 4 − 7 2 ÷ 12 × 3 − 6 4
• ÷ − × 3 2 × 4 3 + 15 × 2 − 6 × 3
Exercises

Evaluate the expressions, which are given in reverse
Polish, or postfix, notation
• 4 3 2 ÷ − 5 × 4 2 × 5 × 3 ÷ ÷
• 3 7 × 4 − 9 × 6 5 × 2 + ÷
Exercises

Draw a binary tree whose preorder search produces
• JBACDIHEGF
• CATSANDDOGS
Draw a binary tree whose postorder search produces
• SEARCHING
• TREEHOUSE
Exercises

Trees (slides)

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