Data Structures and Algorithms IT 2202

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Data Structures and Algorithms (IT 2202)

• Trees

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In previous lessons

• Overview of the syllabus
• Basic concepts of DS A variables, data
  types ADT, data structures
• Lists, Stacks, and Queues

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In this lesson

• Trees
• Define and describe various types of
  trees(general definition, binary and binary
  search trees, balanced trees, AVL trees, multiway
  and multiway search trees, self adjusting trees)
• Tree traversal techniques(pre-order, in-order,
  post-order)
• Implementations
• Sample questions from past papers

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Introduction

• Lists are more flexible than arrays but are still
  linear structures
• Trees let us hierarchically organize a collection
  of data items

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A definition

• A tree is a collection of nodes
• Together with a relation (parenthood) that
  imposes a hierarchical structure on these nodes.
• One of these nodes is called the root.

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A definition

• Trees are formally defined recursively
• A single node by itself is a tree. This node is
  also the root of this tree.
• Let t1, t2, , tk be disjoint trees with roots
  r1, r2, , rk respectively, and let R be another
  node.We can get a new tree by making R the
  parent of the nodes r1, r2, , rk.

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A definition

• In this tree R is the root and t1, t2, , tk are
  subtrees of the root. Nodes r1, r2, , rk are
  called the children of node R.

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Path

• If n1, n2, nk is a sequence of nodes in a tree
  such that ni is the parent of ni1, for 1i k,
  then this sequence is called a path from node n1
  to nk

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Length of a Path

• The length of a path is one less than the number
  of nodes in the path.
• There is a path of length zero from every node to
  itself.

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Ancestors and Descendants

• If there is a path from node a to node b, then a
  is an ancestor of b, and b is a descendant of a.
• Any node is both an ancestor and a descendant of
  itself.

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Ancestors and Descendants

• An ancestor or a descendant of a node, other than
  the node itself, is called a proper ancestor or
  proper descendant, respectively.
• In a tree, the root is the only node with no
  proper ancestors. A node with no proper
  descendants is called a leaf.

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In Brief

• a is the root
• b and d are leaves
• The height of the tree is 2
• c is an ancestor of d
• b is a descendant of a

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Subtrees

• A subtree of a tree is a node, together with all
  its descendants.

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Height of a node/tree

• The height of a node in a tree is the length of a
  longest path from the node to a leaf.
• The height of a tree is the height of the root.

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Depth of a node

• The depth of a node is the length of the unique
  path from the root to that node.

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Order of siblings

• Children of a node (siblings) are usually ordered
  from left-to-right.
• If we wish to explicitly ignore the order of
  children, we shall refer to a tree as an
  unordered tree.

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Order of siblings

• We can extend the left-to-right ordering of
  siblings to compare two nodes that are not
  related by ancestor-descendant relationship.

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Order of siblings

• The relevant rule is that if a and b are
  siblings, and a is to the left of b, then all
  descendants of a re to the left of all
  descendants of b.

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Tree Traversal

• Tree traversal is the process of visiting each
  node in the tree exactly one time.
• Traversing Methods
• Breadth First
• Depth First

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Breadth First Traversal

30, 15, 45, 10, 13, 42, 29
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Depth First Traversal

• Three methods of depth first traversal is defined
• Preorder
• Inorder
• Postorder

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Preorder Tree Traversal

30, 15, 10, 13, 45, 42, 29
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Inorder Tree Traversal

10, 15, 13, 30, 42, 45, 29
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Postorder Tree Traversal

10, 13, 15, 42, 29, 45, 30
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Binary Trees

• A binary tree is either an empty tree, or a tree
  in which every node has either no children, only
  a left child, only a right child, or both a left
  child and a right child.

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A Complete Binary Tree

• A Complete Binary Tree is a binary tree where
• Nodes at all levels, except the last have exactly
  two non-null children.
• The number of nodes
• Number of leaves
• Number of non-leaf nodes
• 2h-1 (2h-1 1 ) 2h - 1
• (h is the height of the tree)

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Binary Search Trees

• A binary search tree is a binary tree where
• For each node n of the tree the values stored in
  the nodes of its left subtree are less than the
  value v stored at that node. And the values
  stored in the nodes of its right subtree are
  greater.
• This is called the binary search tree
  property. Multiple copies of the same value are
  not stored.

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Balanced Trees

• A binary tree is a balanced tree if the
  difference in the heights of both subtrees of any
  node is either 0 or 1.
• A binary tree is perfectly balanced if
• It is balanced
• And all leaves are found at one or two levels

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AVL Trees

• An AVL tree is a tree where the height of the
  left and right subtrees of every node differ by
  at most one.
• The accepted balance factors are
• -1, 0, 1
• AVL Adelson Velskii Landis

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AVL Tree

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Self Adjusting Trees

• Is an alternative method of improving the
  performance of the insert, delete, and retrieve
  operation on a binary tree by
• Moving the most often accessed nodes up the tree

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Multiway Trees

• A tree that can have more than 2 children at its
  nodes. (as opposed to binary trees)
• A multiway search tree is m-tree, where
• Each node has m children and m-1 keys
• The keys in each node are in ascending order
• The keys in the first i children are smaller than
  the ith key
• The keys in the last m-1 children are larger than
  the ith key

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Implementation of Trees

• Commonly implemented
• By Using arrays
• As Linked structures

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Question 1

• Consider the following statements is/are true?
• 1. A binary tree can contain at least 2L nodes at
  level L.
• 2. A complete binary tree of depth d is a binary
  tree that contains 2L Nodes at each level L
  between 0 and d, both inclusive.

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Question 1

• 3. The total number of nodes (Tn ) in a complete
  binary tree of depth d is
• 2 d1 - 1 .
• 4. The height of the complete binary tree can be
  written as h log2 (Tn1)-1 where Tn is Total
  number of Nodes.

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Question 1

• Which of them is correct in respect of the above
  statements regarding the Binary trees?
• A complete binary tree of depth d is a binary
  tree that contains 2L Nodes at each level L
  between 0 and d, both inclusive.
• The total number of nodes (Tn ) in a complete
  binary tree of depth d is 2d1 - 1 .
• The height of the complete binary tree can be
  written as h log2 (Tn1)-1 where Tn is Total
  number of Nodes.

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Question 2

• 30, 12, 17, 49, 22, 65, 51, 56, 70, 68 is a set
  of 10 integers.
• Create a Binary search tree using the above set
  of integers.

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Question 2
30

30, 12, 17, 49, 22, 65, 51, 56, 70, 68
49
12
65
17
51
22
70
56
68
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Question 3

• Which of the following is/are not (an) AVL
  tree(s)?

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Question 3

Not AVL Tree
AVL Tree
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Question 3

Not AVL Tree
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Question 4

• Consider the following expression tree

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Question 4

• Which of the following expressions shows the post
  order traversal of the above tree?
• (a) abcdefg
• (b) abcdefg
• (c) abcdefg
• (d) abcdefg
• (e) abcdefg

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Review of what we have covered

• Define and describe various types of
  trees(general definition, binary and binary
  search trees, balanced trees, AVL trees, multiway
  and multiway search trees, self adjusting trees)
  
• Tree traversal techniques(pre-order, in-order,
  post-order)
• Implementations
• A sample question from a past paper

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In the next lesson

• Graphs