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Why Trees-Computer Programming For Aeronautical Engineering And Sciences-Lecture Slides, Slides of Aeronautical Engineering

University of AllahabadAeronautical Engineering

Prof. Balamohan Pawar delivered this lecture at Allahabad University for Aeronautical Engineering and Computer Programming course. Its main points are: Theorem, Lemma, Colorrary, Claim, Proposition, Rooted, Graph, Tree, Vertex, Internal, Property, Binary

Typology: Slides

2011/2012

Uploaded on 07/20/2012

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[Theorem]
•Theorem: a mathematical statement that
can be shown to be true
– Can be proved using other theorems, axioms
(statements which are given to be true) and rules
of inference
•Lemma: a pre-theorem or result needed to
prove a theorem
•Corollary: post-theorem or result which
follows directly from a theorem
• Proposition
•Claim
•Remark
Why should we use trees?
4 2 15 9 5 7
5
4
2
9
7 15
Binary Search Tree
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Download Why Trees-Computer Programming For Aeronautical Engineering And Sciences-Lecture Slides and more Slides Aeronautical Engineering in PDF only on Docsity!

[Theorem]

  • Theorem : a mathematical statement that can be shown to be true - Can be proved using other theorems, axioms (statements which are given to be true) and rules of inference
  • Lemma : a pre-theorem or result needed to prove a theorem
  • Corollary : post-theorem or result which follows directly from a theorem
  • Proposition
  • Claim
  • Remark

Why should we use trees?

Binary Search Tree

5

A tree is a connected undirected graph

with no simple circuits.

  • it cannot contain multiple edges or loops

Trees

Theorem : An undirected graph is a tree if and only if there is a unique simple path between any two of its vertices.

6

Which graphs are trees?

a) b)

c) d)

9

• A vertex that has children is called an

internal vertex

• A graph H( W, F ) is a subgraph of a

graph G ( V,E ) iff W ⊆ V and F ⊆ E

• The subtree at vertex v is the subgraph

of the tree consisting of vertex v and its

descendants and all edges incident to

those descendants

Internal Vertex

10

  • The parent of a non-root vertex v is the unique vertex u with a directed edge from u to v.
  • A vertex is called a leaf if it has no children.
  • The ancestors of a non-root vertex are all the vertices in the path from root to this vertex.
  • The descendants of vertex v are all the vertices that have v as an ancestor.

Tree Properties

11

• The level of vertex v in a rooted tree is the

length of the unique path from the root to v.

• The height of a rooted tree is the maximum

of the levels of its vertices.

Tree Properties

c

a

b

d e f g

k l

m

h i j

Level of vertex f = Height of tree =

Level of vertex f = Height of tree =

2 4

12

• An m-ary tree is a rooted tree in which

each internal vertex has at most m children

• A rooted tree is called a binary tree if

every internal vertex has no more than 2

children.

• The tree is called a full binary tree if every

internal vertex has exactly 2 children.

Binary Tree

15

• 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.

Ordered Binary Tree

Example

Children of b? Parent of b? Ancestors of g? Descendants of b?

Leafs? Internal vertices? Left child of g? Right child of g?

c

a

b

d e f g

k l

m

h i j

d, e a c, a

FAREWELL TO AMY ADAMS The party atmosphere of Motown and theterrific backing rhythms of the Funk Brothers just couldn't bring out the best in Amy, whowas sent home this week.

d, e, h, i

h, i, e, j, k, m a, b, c, d, f, g k l

17

  • A traversal algorithm is a procedure for systematically visiting every vertex of an ordered binary tree
  • Tree traversals are defined recursively
  • Three commonly used traversals are:
    • preorder
    • inorder
    • postorder

Traversal Algorithms

18

Let T be an ordered binary tree with root R

If T has only R then R is the preorder traversal Else Let T 1 , T 2 be the left and right subtrees at R Visit R Traverse T 1 in preorder Traverse T 2 in preorder

PREORDER Traversal Algorithm

21

POSTORDER Traversal Algorithm

Let T be an ordered binary tree with root R

If T has only R then R is the postorder traversal Else Let T 1 , T 2 be the left and right subtrees at R Traverse T 1 in postorder Traverse T 2 in postorder Visit R

22

A special kind of binary tree in which:

  • Each leaf node contains a single operand
  • Each inner vertex contains a single binary operator
  • The left and right subtrees of an operator node represent sub-expressions that must be evaluated before applying the operator at the root of the subtree.

Binary Expression Tree

23

Binary Expression Tree

ROOT

INORDER TRAVERSAL: 8 - 5 has value 3 PREORDER TRAVERSAL: - 8 5 POSTORDER TRAVERSAL: 8 5 -

24

Binary Expression Tree

What value does it have?What value does it have?

( 4 + 2 ) * 3 = 18

27

Binary Expression Tree

**Infix: ( ( 8 - 5 ) * ( ( 4 + 2 ) / 3 ) ) Prefix: * - 8 5 / + 4 2 3 Postfix: 8 5 - 4 2 + 3 / ***

Trees - Glossary

Perfectly balanced tree Height balanced tree

Root

Leaf

Inner Vertex

A

B C

Parent of B and C

Child of A

B and C are siblings

M-ary tree