Transportation Modelling - Quantitative Analysis and Discrete Maths
Discrete Maths Trees
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Transcript of 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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MATH 224 Discrete Mathematics
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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MATH 224 Discrete Mathematics
A Binary Search Tree
15
3010
535
32 458
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MATH 224 Discrete Mathematics
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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MATH 224 Discrete Mathematics
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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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 j
T is NULL
T is d
prints d
T is b
T is a
main calls
inorder
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Binary Tree Traversal Functions
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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Binary Tree Traversal Functions
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?