Trees

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Trees

• EENG212
• Algorithms
• and
• Data Structures

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Trees

• Outline
• Introduction to Trees
• Binary Trees Basic Definitions
• Traversing Binary Trees
• Node Representation of Binary Trees
• Primitive Functions in Binary Trees

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Introduction to Trees

• BINARY TREES BASIC DEFINITIONS
• A binary tree is a finite set of elements that
  are either empty or is partitioned into three
  disjoint subsets. The first subset contains a
  single element called the root of the tree. The
  other two subsets are themselves binary trees
  called the left and right subtrees of the
  original tree. A left or right subtree can be
  empty.
• Each element of a binary tree is called a node of
  the tree. The following figure shows a binary
  tree with 9 nodes where A is the root.

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BINARY TREES BASIC DEFINITIONS
root
left subtree
right subtree
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BINARY TREES BASIC DEFINITIONS

• If A is the root of a binary tree and B is the
  root of its left or right subtrees, then A is
  said to be the father of B and B is said to be
  the left son of A.
• A node that has no sons is called the leaf.
• Node n1 is the ancestor of node n2 if n1 is
  either the father of n2 or the father of some
  ancestor of n2. In such a case n2 is a descendant
  of n1.
• Two nodes are brothers if they are left and right
  sons of the same father.

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BINARY TREES BASIC DEFINITIONS
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BINARY TREES BASIC DEFINITIONS

• If every nonleaf node in a binary tree has
  nonempty left and right subtrees, the tree is
  called a strictly binary tree.

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BINARY TREES BASIC DEFINITIONS

• The level of a node in a binary tree is defined
  as follows The root of the tree has level 0, and
  the level of any other node in the tree is one
  more than the level of its father.
• The depth of a binary tree is the maximum level
  of any leaf in the tree.
• A complete binary tree of depth d is the strictly
  binary all of whose leaves are at level d. A
  complete binary tree with depth d has 2d leaves
  and 2d-1 nonleaf nodes.

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BINARY TREES BASIC DEFINITIONS
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TRAVERSING BINARY TREES

• One of the common operations of a binary tree is
  to traverse the tree. Traversing a tree is to
  pass through all of its nodes once. You may want
  to print the contents of each node or to process
  the contents of the nodes. In either case each
  node of the tree is visited.
• There are three main traversal methods where
  traversing a binary tree involves visiting the
  root and traversing its left and right subtrees.
  The only difference among these three methods is
  the order in which these three operations are
  performed.

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TRAVERSING BINARY TREES

• Traversing a binary tree in preorder (depth-first
  order)
• 1. Visit the root.
• 2. Traverse the left subtree in preorder.
• 3. Traverse the right subtree in preorder.

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Traversing a binary tree in preorder
Preorder ABDGCEHIF
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TRAVERSING BINARY TREES

• Traversing a binary tree in inorder
• (or symmetric order)
• 1. Traverse the left subtree in inorder.
• 2. Visit the root.
• 3. Traverse the right subtree in inorder.

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Traversing a binary tree in inorder
Inorder DGBAHEICF
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TRAVERSING BINARY TREES

• Traversing a binary tree in postorder
• 1. Traverse the left subtree in postorder.
• 2. Traverse the right subtree in postorder.
• 3. Visit the root.

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Traversing a binary tree in postorder
Postorder GDBHIEFCA
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NODE REPRESENTATION OF BINARY TREES

• Each node in a binary tree contains info, left,
  right and father fields. The left, right and
  father fields points the nodes left son, right
  son and the father respectively.
• struct node
• int info / can be of different type/
• struct node left
• struct node right
• struct node father
• typedef struct node NODEPTR

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PRIMITIVE FUNCTIONS IN BINARY TREES

• The maketree function allocates a node and sets
  it as the root of a single node binary tree.
• NODEPTR maketree(int x)
• NODEPTR p
• p getnode()
• p-gtinfo x
• p-gtleft NULL
• p-gtright NULL
• return p

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PRIMITIVE FUNCTIONS IN BINARY TREES

• The setleft and setright functions sets a node
  with content x as the left son and right son of
  the node p respectively.

void setleft(NODEPTR p, int x) if(p NULL) printf(void insertion\n) else if (p-gtleft ! NULL) printf(invalid insertion\n) else p-gtleft maketree(x) void setright(NODEPTR p, int x) if(p NULL) printf(void insertion\n) else if (p-gtright ! NULL) printf(invalid insertion\n) else p-gtright maketree(x)
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BINARY TREE TRAVERSAL METHODS

• Recursive functions can be used to perform
  traversal on a given binary tree. Assume that
  dynamic node representation is used for a given
  binary tree.
• In the following traversal methods, the tree is
  traversed always in downward directions.
  Therefore the father field is not needed.
• The following recursive preorder traversal
  function displays the info part of the nodes in
  preorder. Note that the info part is integer
  number and tree is a pointer to the root of the
  tree.

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BINARY TREE TRAVERSAL METHODS

• void pretrav(NODEPTR tree)
• if(tree ! NULL)
• printf(d\n, tree-gtinfo)
• pretrav(tree-gtleft)
• pretrav(tree-gtright)

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BINARY TREE TRAVERSAL METHODS

• The following recursive inorder traversal
  function displays the info part of the nodes in
  inorder.
• Note that the info part is integer number and
  tree is a pointer to the root of the tree.

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BINARY TREE TRAVERSAL METHODS

• void intrav(NODEPTR tree)
• if(tree ! NULL)
• intrav(tree-gtleft)
• printf(d\n, tree-gtinfo)
• intrav(tree-gtright)

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BINARY TREE TRAVERSAL METHODS

• The following recursive postorder traversal
  function displays the info part of the nodes in
  postorder.
• Note that the info part is integer number and
  tree is a pointer to the root of the tree.

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BINARY TREE TRAVERSAL METHODS

• void posttrav(NODEPTR tree)
• if(tree ! NULL)
• posttrav(tree-gtleft)
• posttrav(tree-gtright)
• printf(d\n, tree-gtinfo)

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BINARY SEARCH TREE AN APPLICATION OF BINARY TREES

• A binary tree, that has the property that all
  elements in the left subtree of a node n are less
  than the contents of n, and all elements in the
  right subtree of n are greater than or equal to
  the contents of n, is called a Binary Search Tree
  or Ordered Binary Tree.

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BINARY SEARCH TREE AN APPLICATION OF BINARY TREES

• Given the following sequence of numbers,
• 14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14,
  5
• The following binary search tree can be
  constructed.

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BINARY SEARCH TREE AN APPLICATION OF BINARY TREES
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BINARY SEARCH TREE AN APPLICATION OF BINARY TREES

• The inorder (left-root-right) traversal of the
  above Binary Search Tree and printing the info
  part of the nodes gives the sorted sequence in
  ascending order. Therefore, the Binary search
  tree approach can easily be used to sort a given
  array of numbers.
• The inorder traversal on the above Binary Search
  Tree is
• 3, 4, 4, 5, 5, 7, 9, 9, 14, 14, 15, 16, 17, 18,
  20

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SEARCHING THROUGH THE BINARY SEARCH TREE

• Searching operation of the binary search tree is
  always in downward direction. Therefore the
  following node structure can be used to represent
  the node of a given binary search tree.
• Note that the father link is not required.

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SEARCHING THROUGH THE BINARY SEARCH TREE

• struct node
• int info / can be of different type/
• struct node left
• struct node right
• typedef struct node NODEPTR

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SEARCHING THROUGH THE BINARY SEARCH TREE

• The following recursive function can be used to
  search for a given key element in a given array
  of integers. The array elements are stored in a
  binary search tree. Note that the function
  returns TRUE (1) if the searched key is a member
  of the array and FALSE (0) if the searched key is
  not a member of the array.

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SEARCHING THROUGH THE BINARY SEARCH TREE

• int BinSearch(NODEPTR p, int key)
• if(p NULL)
• return FALSE
• else
• if (key p-gtinfo)
• return TRUE
• else
• if(key lt p-gtinfo)
• return BinSearch(p-gtleft, key)
• else
• return BinSearch(p-gtright, key)

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INSERTING NODES INTO A BINARY SEARCH TREE

• The following recursive function can be used to
  insert a new node into a given binary search
  tree.

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NODEPTR insert(NODEPTR p, int x) if(p
NULL) p getnode() p-gtinfo x p-gtleft
NULL p-gtright NULL return p else if(x lt
p-gtinfo) p-gtleft insert(p-gtleft,
x) else p-gtright insert(p-gtright, x) return
p
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Application of Binary Search Tree

• Suppose that we wanted to find all duplicates in
  a list of numbers. One way of doing this to
  compare each number with all those that precede
  it. However this involves a large number of
  comparison. The number of comparison can be
  reduced by using a binary tree. The first number
  in the list is placed in a node that is the root
  of the binary tree with empty left and right
  sub-trees. The other numbers in the list is than
  compared to the number in the root. If it is
  matches, we have duplicate. If it is smaller, we
  examine the left sub-tree if it is larger we
  examine the right sub-tree. If the sub-tree is
  empty, the number is not a duplicate and is
  placed into a new node at that position in the
  tree. If the sub-tree is nonempty, we compare the
  number to the contents of the root of the
  sub-tree and the entire process is repeated with
  the sub-tree. A program for doing this follows .

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include ltstdio.hgt include ltstdlib.hgt struct
node struct node left int info
struct node right typedef struct node
NODEPTR NODEPTR maketree(int) NODEPTR
getnode(void) void intrav(NODEPTR) void
main() int number NODEPTR root , p , q
printf("s\n","Enter First number")
scanf("d",number) rootmaketree(number) /
insert first root item / printf("s\n","Enter
the other numbers")
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while(scanf("d",number) !EOF) pqroot
/ find insertion point / while((number
!p-gtinfo) q!NULL) pq if
(number ltp-gtinfo) q p-gtleft
else q p-gtright
qmaketree(number) / insertion / if
(numberp-gtinfo) printf("d is a
duplicate \n",number) else if
(numberltp-gtinfo)
p-gtleftq else p-gtrightq
printf("Tree Created \n ") / inorder
Traversing / intrav(root)
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void intrav(NODEPTR tree) if(tree !
NULL) intrav(tree-gtleft) printf(d\n,
tree-gtinfo) intrav(tree-gtright) NODEPTR
maketree(int x) NODEPTR p p
getnode() p-gtinfo x p-gtleft NULL p-gtright
NULL return p NODEPTR getnode(void)
NODEPTR p p(NODEPTR) malloc(sizeof(struct
node)) return p