discrete maths trees

8/22/2019 Discrete Maths Trees

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MATH 224 Discrete Mathematics

A rooted tree is a tree that has a distinguished node called the root. Just as with all

trees a rooted tree is acyclic (no cycles) and connected. Rooted trees have many

applications in computer science, for example the binary search tree.

Rooted Trees

The root is typically drawn at the top.

Children of the root.

Grandchildren of the root and leaf nodes.


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A Binary tree is a rooted tree where no node has more than two children. As shown in

the previous slide all nodes, except for leaf nodes, have children and some have

grandchildren. All nodes except the root have a parent and some have a grandparent

and ancestors. A node has at most one parent and at most one grandparent.

Rooted Binary Trees

The root of a binary tree at level 4 also height 4 and depth 0.

Nodes with only 1 child at level 2 and depth 2.

Nodes a level 1, and depth 3.

Nodes at level 3 and depth 1.

Nodes with no childrenare called leaf nodes.

Nodes a level 0, and depth 4.


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A Balanced tree is a rooted tree where the leaf nodes have depths that vary by no

more than one. In other words, the depth of a leaf node is either equal to the height of

the tree or one less than the height. All of the trees below are balanced.

Balanced Trees

What are the heights of each of these trees?


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A Binary Search Tree

15

3010

535

32 458


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Some Tree Applications

Binary Search Trees to store items for easy retrieval, insertion and deletion. For

balanced trees, each of these steps takes lg(N) time for an Nnode tree.

Variations of the binary search tree include, the AVL tree, Red-Blacktree and the

B-tree. These are all designed to keep the tree nearly balanced. The first two are

binary trees while the B-tree has more than two children for each internal node.Game trees are used extensively in AI. The text has a simple example of a tic-tac-

toe tree on Page 654.

Huffman trees are used to compress data. They are most commonly used to

compress faxes before transmission.

Spanning trees are subgraphs that have applications to computer and telephonenetworks. Minimum spanning trees are of special interest.

Steiner trees are a generalization of the spanning tree with in multicast

communication.


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Binary Tree Traversal Functions

void inorder(node T)

if (T null) {

inorder(T>left)

cout data right)

} fiend inorder

a

e

c

d

b

f g

h

i joutput will be something like

d, b, e, a, f, c, g, i, h, j


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void inorder(node T)

if (T null) {

inorder(T>left)

cout data right)

} fiend inorder

a

e

c

d

b

f g

h

i j

T is NULL

T is d

prints d

T is b

T is a

main calls

inorder


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void pre_order(node T)

if (T null) {

cout data left)

pre_order(T >right)

}end pre_order

a

e

c

d

b

f g

h

i joutput will be something like

a, b, d, e, c, f, g, h, i, j


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void post_order(node T)

if (T null) {

post_order(T>left)

post_order(T >right)

cout data


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Counting the nodes in a Binary Tree

int count(node T)

int num = 0

if (T null)

num = 1 + count(T>left) + count(T >right)

fi

return numend count

a

e

c

d

b

f g

h

i j


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The Height of a Binary Tree

int height(node T)int ht = 1

if (T null)

ht = 1 + max(height(T>left), height(T>right))

fi

return ht

end height

a

e

c

d

b

f g

h

i j

Called initially at a

Then at b

Then a d

Then at NULL

Returns 1 to dfrom left and right

Returns 0 to bfrom dand from e

Returns 1 to afrom left.

Returns 3 to afrom right

Returns 4 from ato original caller


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Inserting into a Binary Search Tree

// Note that T is passed by referenceinsert(node & T, int X)

if (T = = null)

T = node(X)

else if X < T>value

insert(T >left, X)

elseinsert(T >right, X)

fi

end height

30

24

45

17

22

32 70

84

75 98

Where would 60 be inserted?

If the original value for T is the

node containing 30, where is the

first recursive call, the second etc?

Where would 10 be inserted?

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