CSCI201 DATA STRUCTURES Data and Program Organization

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CSCI201 DATA STRUCTURES(Data and Program
Organization)
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• Binary Tree
• Properties and Representation

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Definition of a Binary Tree

• A binary tree is a tree in which no node can
  have more than two children. Because there are
  only two children we name them left and right.

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Recursive Definition of a Binary Tree

• A binary tree is either empty or consists of a
  root, a left tree and a right tree. The left and
  right trees may themselves be empty. Thus a node
  with one child could have a left or a right child.

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An expression tree such as this one is an
essential data structure in compiler design.
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Binary Tree Properties Representation
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Minimum Number Of Nodes in a Binary Tree
minimum number of nodes of a binary tree of
height h is h 1
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Maximum Number Of Nodes
Maximum number of nodes (N) in a binary tree is
2h1 - 1 For a height h of 3, N 1 2 4 8
15 ( 24 - 1)
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Number Of Nodes Height

• Let n be the number of nodes in a binary tree
  whose height is h.
• h1 lt n lt 2h1 1
• i.e. the value of n lies between the minimum
  h1 and the maximum 2h1 1
• log2(n1) lt h1 lt n
• e.g. for n 7, then log2(8) height of 2
  for a balanced binary tree, or height 6 for
  binary tree where each node has only one child .
  . .

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The Logarithm

• 2x n
• 23 8
• log28 3
• When logarithms are used in computer science,
  the base is always 2. Note, the log base 2 of a
  number always results in a smaller number.

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Balanced complete binary tree. n 7, h 2
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Unbalanced complete binary tree. n 6, h 2
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Unbalanced incomplete binary tree. n 6, h 2
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Right-skewed binary tree. n 7, h 6
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Full Binary Tree

• A full binary tree of a given height h has 2h1
  1 nodes.

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Numbering Nodes In A Full Binary Tree

• Number the nodes 1 through 2h1 1.
• Number by levels from top to bottom.
• Within a level number from left to right.

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Node Number Properties

• Parent of node i is node i / 2, unless i 1.
• Node 1 is the root and has no parent.

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Node Number Properties

• Left child of node i is node 2i, unless 2i gt n,
  where n is the number of nodes.
• If 2i gt n, node i has no left child.

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Node Number Properties

• Right child of node i is node 2i1, unless 2i1 gt
  n, where n is the number of nodes.
• If 2i1 gt n, node i has no right child.

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Complete Binary Tree With n Nodes

• Start with a full binary tree that has at least n
  nodes.
• Number the nodes as described earlier.
• The binary tree defined by the nodes numbered 1
  through n is the unique n node complete binary
  tree.

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Example

• Complete binary tree with 10 nodes.

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Example

• Complete binary tree with 15 nodes.

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Binary Tree Representation

• Array representation.
• Linked representation.

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Array Representation

• Number the nodes using the numbering scheme for a
  full binary tree. The node that is numbered i is
  stored in treei.

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b
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j
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Right-Skewed Binary Tree

• An n node binary tree needs an array whose length
  is between n1 and 2n.

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Linked Representation

• Each binary tree node is represented as an object
  whose data type is BinaryTreeNode.
• The space required by an n node binary tree is n
  (space required by one node).

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The Class BinaryTreeNode

• package dataStructures
• public class BinaryTreeNode
• Object element
• BinaryTreeNode leftChild // left subtree
• BinaryTreeNode rightChild// right subtree
• // constructors and any other methods
• // come here

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Linked Representation Example
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Some Binary Tree Operations

• Determine the height.
• Determine the number of nodes.
• Make a clone.
• Determine if two binary trees are clones.
• Display the binary tree.
• Evaluate the arithmetic expression represented by
  a binary tree.
• Obtain the infix form of an expression.
• Obtain the prefix form of an expression.
• Obtain the postfix form of an expression.

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Binary Tree Traversal

• Many binary tree operations are done by
  performing a traversal of the binary tree.
• In a traversal, each element of the binary tree
  is visited exactly once.
• During the visit of an element, all action (make
  a clone, display, evaluate the operator, etc.)
  with respect to this element is taken.

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Binary Tree Traversal Methods

• Preorder
• Inorder
• Postorder
• Level order

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• End of BinaryTrees.ppt

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• Last Updated Friday 1st October 2004, 954 PT,
  AHD