General Tree Concepts Binary Trees

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General Tree ConceptsBinary Trees
Mark Allen Weiss Data Structures and Algorithm
Analysis in Java

• Data Structures and Algorithms

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Outline

• Tree Data Structure
• Examples
• Definition
• Terminology
• Binary Tree
• Definitions
• Examples
• Tree Traversal
• Binary Search Trees
• Definition

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Linear Lists and Trees

• Linear lists (arrays, linked list)are useful for
  serially ordered data
• (e1,e2,e3,,en)
• Days of week
• Months in a year
• Students in a class
• Trees are useful for hierarchically ordered data
• Khalids descendants
• Corporate structure
• Government Subdivisions
• Software structure

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Examples of trees

• directory structure
• family trees
• all descendants of a particular person
• all ancestors born after year 1800 of a
  particular person
• evolutionary tress (also called phylogenetic
  trees)
• albegraic expressions

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Khalids Descendants
What are other examples of hierarchically ordered
data?
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Property of a Tree

• A tree t is a finite nonempty set of nodes
• One of these elements is called the root
• Each node is connected by an edge to some other
  node
• A tree is a connected graph
• There is a path to every node in the tree
• There are no cycles in the tree.
• It can be proved that a tree has one less edge
  than the number of nodes.
• Usually depicted with the root at the top

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Is it a Tree?
NO! All the nodes are not connected
yes! (but not a binary tree)
yes!
NO! There is a cycle and an extra edge (5 nodes
and 5 edges)
yes! (its actually the same graph as the blue
one) but usually we draw tree by its levels
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Subtrees
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Tree Terminology

• The element at the top of the hierarchy is the
  root.
• Elements next in the hierarchy are the children
  of the root.
• Elements next in the hierarchy are the
  grandchildren of the root, and so on.
• Elements at the lowest level of the hierarchy are
  the leaves.

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Other Definitions

• Leaves, Parent, Grandparent, Siblings,Ancestors,
  Descendents

Leaves Hassan,Gafar,Sami,Mohmmed
Parent(Musa) Khalid
Grandparent(Sami) Musa
Siblings(Musa) Ahmed,Omer
Ancestors(Hassan) Amed,Khalid
Descendents(Musa)Fahad,Sami
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Levels and Height

• Root is at level 0 and its children are at level
  1.

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Degree of a node

• Node degree is the number of children it has

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Tree Degree

• Tree degree is the maximum of node degrees

tree degree 3
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Measuring Trees

• The height of a node v is the number of nodes on
  the longest path from v to a leaf
• The height of the tree is the height of the root,
  which is the number of nodes on the longest path
  from the root to a leaf
• The depth of a node v is the number of nodes on
  the path from the root to v
• This is also referred to as the level of a node
• Note that there are slightly different
  formulations of the height of a tree
• Where the height of a tree is said to be the
  length (the number of edge) on the longest path
  from node to a leaf

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Trees - Example
Level
0
1
2
3
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Tree Terminology Example

• A is the root node
• B is the parent of D and E
• C is the sibling of B
• D and E are the children of B
• D, E, F, G, I are external nodes, or leaves
• A, B, C, H are internal nodes
• The depth, level, or path length of E is 2
• The height of the tree is 3
• The degree of node B is 2

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Tree Terminology Example
A
path from A to D to G
B
D
subtree rooted at B
leaves C,E,F,G
G
E
F
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Tree Representation
Class TreeNode Object element
TreeNode firstChild TreeNode
nextSibling
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Binary Trees

• A finite (possibly empty) collection of elements
• A nonempty binary tree has a root element and the
  remaining elements (if any) are partitioned into
  two binary trees
• They are called the left and right subtrees of
  the binary tree

A
B
C
right child of A
left subtree of A
F
G
D
E
right subtree of C
H
I
J
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Difference Between a Tree a Binary Tree

• A binary tree may be empty a tree cannot be
  empty.
• No node in a binary tree may have a degree more
  than 2, whereas there is no limit on the degree
  of a node in a tree.
• The subtrees of a binary tree are ordered those
  of a tree are not ordered.

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Binary Tree for Expressions
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Height of a Complete Binary Tree
At each level the number of the nodes is doubled.
total number of nodes
1 2 22 23 24 - 1 15
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Nodes and Levels in a Complete Binary Tree
Number of the nodes in a tree with M levels 1
2 22 . 2M 2 (M1) - 1 22M - 1 Let
N be the number of the nodes. N 22M - 1,
22M N 1 2M (N1)/2 M log( (N1)/2
) N nodes log( (N1)/2 ) O(log(N))
levels M levels 2 (M1) - 1 O(2M )
nodes
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Binary Tree Properties

1. The drawing of every binary tree with n elements,
   n gt 0, has exactly n-1 edges.
2. A binary tree of height h, h gt 0, has at least h
   and at most 2h-1 elements in it.

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Binary Tree Properties

• 3. The height of a binary tree that contains n
  elements, n gt 0, is at least ?(log2(n1))? and
  at most n.

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Full Binary Tree

• A full binary tree of height h has exactly 2h-1
  nodes.
• Numbering the nodes in a full binary tree
• Number the nodes 1 through 2h-1
• Number by levels from top to bottom
• Within a level, number from left to right

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Node Number Property of Full Binary Tree

• 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 Property of Full Binary Tree

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

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Node Number Property of Full Binary Tree

• Right child of node i is node 2i1, unless 2i1 gt
  n,where n is the total 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 n-node complete binary tree.
• A full binary tree is a special case of a
  complete binary tree

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Example of Complete Binary Tree

• Complete binary tree with 10 nodes.
• Same node number properties (as in full binary
  tree) also hold here.

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Binary Tree Representation

• Array representation
• Linked representation

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Array Representation of Binary Tree

• The binary tree is represented in an array by
  storing each element at the array position
  corresponding to the number assigned to it.

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Incomplete Binary Trees

• Complete binary tree with some missing elements

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Example Binary Tree
To find a nodes parent use this
formula  Parents Index (Childs Index 1) /
2
Array index from 0 Ex. To find nodes 2 left and
right child Left Childs Index 2 Parents
Index 1 Right Childs Index 2 Parents
Index 2
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Linked Representation of Binary Tree

• The most popular way to present a binary tree
• Each element is represented by a node that has
  two link fields (leftChild and rightChild) plus
  an element field
• Each binary tree node is represented as an object
  whose data type is binaryTreeNode
• The space required by an n node binary tree isn
  sizeof(binaryTreeNode)

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Linked Representation of Binary Tree
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Node Class For Linked Binary Tree

• TreeNode class
• private class TreeNode
• private String data
• private TreeNode left
• private TreeNode right
•  
• public TreeNode(String element)
• data element
• left null
• right null

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Common Binary Tree Operations

• Determine the height
• Determine the number of nodes
• Make a copy
• Determine if two binary trees are identical
• Display the binary tree
• Delete a tree
• If it is an expression tree, evaluate the
  expression
• If it is an expression tree, obtain the
  parenthesized form of the expression

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Binary Tree Traversal

• A traversal of a tree is a systematic way of
  accessing or "visiting" all the nodes in the
  tree.
• There are three basic traversal schemes
  preorder, postorder, inorder
• .In any of these traversals we always visit the
  left subtree before the right

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Binary Tree Traversal Methods

• Preorder
• The root of the subtree is processed first before
  going into the left then right subtree (root,
  left, right).
• Inorder
• After the complete processing of the left subtree
  the root is processed followed by the processing
  of the complete right subtree (left, root,
  right).
• Postorder
• The root is processed only after the complete
  processing of the left and right subtree (left,
  right, root).
• Level order
• The tree is processed by levels. So first all
  nodes on level i are processed from left to right
  before the first node of level i1 is visited

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Preorder Traversal

• public void preOrderPrint()
• preOrderPrint(root)
•   private void preOrderPrint(TreeNode tree)
• if (tree ! null)
• System.out.print(tree.data " ") //
  Visit the root
• preOrderPrint(tree.left)// Traverse
  the left subtree
• preOrderPrint(tree.right) // Traverse
  the right subtree
•  

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Preorder Example (visit print)
N-L-R
O M N K I J L G E F C A B D H
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Preorder of Expression Tree

• / a b - c d e f
• Gives prefix form of expression.

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Inorder Traversal
public void inOrderPrint()
inOrderPrint(root)   public void
inOrderPrint(TreeNode t) if (t ! null)
inOrderPrint(t.left)
System.out.print(t.data " ")
inOrderPrint(t.right)
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Inorder Example (visit print)
O N M L K J I H G F E D C B A
L-N-R
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Inorder by Projection
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Inorder of Expression Tree

• Gives infix form of expression, which is how we
  normally write math expressions. What about
  parentheses?

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Postorder Traversal
public void ostOrderPrint()
postOrderPrint(root)   public void
postOrderPrint(TreeNode t) if (t ! null)
postOrderPrint(t.left)
postOrderPrint(t.right) System.out.print(t.
data " ")
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Postorder Example (visit print)
L-R-N
H L N O M J K I D F G E B C A
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Postorder of Expression Tree

• a b c d - e f /
• Gives postfix form of expression.

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Level Order Example (visit print)

• Add and delete nodes from a queue
• Output a b c d e f g h i j

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Space and Time Complexity

• The space complexity of each of the four
  traversal algorithm is O(n)
• The time complexity of each of the four traversal
  algorithm is O(n)

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Summary

• Tree, Binary Tree
• In order to process the elements of a tree, we
  consider accessing the elements in certain order
• Tree traversal is a tree operation that involves
  "visiting (or" processing") all the nodes in a
  tree.
• Types of traversing
• Pre-order Visit node first, pre-order all its
  subtrees from leftmost to rightmost.
• Inorder Inorder the node in left subtree and
  then visit the root following by inorder
  traversal of all its right subtrees.
• Post-order Post-order the node in left subtree
  and then post-order the right subtrees followed
  by visit to the node.