Lecture 17: Trees and Networks I

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Lecture 17 Trees and Networks I

• Discrete Mathematical Structures
• Theory and Applications

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Learning Objectives

• Learn the basic properties of trees
• Explore applications of trees
• Learn about networks

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Trees

• Each atom of a chemical compound is represented
  by a point in a plane
• Atomic bonds are represented by lines
• Shown in Figure 11.1 for the chemical compound
  with the formula C4H10.

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Trees

• In chemistry, chemical compounds with formula
  CkH2k2 are known as paraffins, which contain k
  carbon atoms and 2k 2 hydrogen atoms.

• In the graphical representation, each of the
  carbon atoms corresponds to a vertex of degree 4
  and each of the hydrogen atoms corresponds to a
  vertex of degree 1.
• For the same chemical formula C4H10, the graph
  shown in Figure 11.2 is also a representation.

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Trees

• These graphs are connected and have no cycles.
  Hence, each of these graphs is a tree.

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Trees

• Consider the graphs shown in Figure 11.4. Each of
  these graphs is connected.
• However, each of these graphs has a cycle. Hence,
  none of these graphs is a tree.

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Trees
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Trees
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Trees
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Rooted Tree
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Rooted Tree

• The level of a vertex v is the length of the path
  from the root to v.

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• The root of this binary tree is A. Vertex B is
  the left child of A and vertex C is the right
  child of A. From the diagram, it follows that B
  is the root of the left subtree of A, i.e., the
  left subtree of the root. Similarly, C is the
  root of the right subtree of A, i.e., the right
  subtree of the root. LA B, D, E, G and RA
  C, F ,H. Moreover, for vertex F , the left
  child is H and F has no right child.

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Rooted Tree
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Rooted Tree

• Binary Tree Traversal
• Item insertion, deletion, and lookup operations
  require the binary tree to be traversed. Thus,
  the most common operation performed on a binary
  tree is to traverse the binary tree, or visit
  each vertex of the binary tree. The traversal
  must start at the root because one is typically
  given a reference to the root. For each vertex,
  there are two choices.
• Visit the vertex first.
• Visit the subtrees first.

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Rooted Tree

• Inorder Traversal In an inorder traversal, the
  binary tree is traversed as follows.
• Traverse the left subtree.
• Visit the vertex.
• Traverse the right subtree.

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Rooted Tree

• Preorder Traversal In a preorder traversal, the
  binary tree is traversed as follows.
• Visit the vertex.
• Traverse the left subtree.
• Traverse the right subtree.

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Rooted Tree

• Postorder Traversal In a postorder traversal,
  the binary tree is traversed as follows.
• Traverse the left subtree.
• Traverse the right subtree.
• Visit the vertex.

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Rooted Tree

• Each of these traversal algorithms is recursive.
• The listing of the vertices produced by the
  inorder traversal of a binary tree is called the
  inorder sequence.
• The listing of the vertices produced by the
  preorder traversal of a binary tree is called the
  preorder sequence.
• The listing of the vertices produced by the
  postorder traversal of a binary tree is called
  the postorder sequence.

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Rooted Tree

• Binary Search Trees
• To determine whether 50 is in the binary tree,
  any of the previous traversal algorithms to visit
  each vertex and compare the search item with the
  data stored in the vertex can be used.
• However, this could require traversal of a large
  part of the binary tree, so the search would be
  slow.
• Each vertex in the binary tree must be visited
  until either the item is found or the entire
  binary tree has been traversed because no
  criteria exist to guide the search.

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Rooted Tree

• Binary Search Trees
• In the binary tree in Figure 11.22, the value of
  each vertex is larger than the values of the
  vertices in its left subtree and smaller than the
  values of the vertices in its right subtree.
• The binary tree in Figure 11.22 is a special type
  of binary tree, called a binary search tree.

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