Trees, Binary Trees, and Binary Search Trees by HC11121317173

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  Trees, Binary Trees,
and Binary Search Trees
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                            Trees
     Linear   access time of linked lists is prohibitive
         Does there exist any simple data structure for
          which the running time of most operations (search,
          insert, delete) is O(log N)?
     Trees
       Basic concepts
       Tree traversal
       Binary tree
       Binary search tree and its operations
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                              Trees
    A   tree T is a collection of nodes
      T can be empty
      (recursive definition) If not empty, a tree T consists
       of
           a (distinguished) node r (the root),
           and zero or more nonempty subtrees T1, T2, ...., Tk
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       Tree can be viewed as a „nested‟ lists
       Tree is also a graph …
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                  Some Terminologies




       Child and Parent
           Every node except the root has one parent
           A node can have an zero or more children
       Leaves
           Leaves are nodes with no children
       Sibling
           nodes with same parent
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                   More Terminologies
       Path
            A sequence of edges
       Length of a path
            number of edges on the path
       Depth of a node
            length of the unique path from the root to that node
       Height of a node
            length of the longest path from that node to a leaf
            all leaves are at height 0
       The height of a tree = the height of the root
                                   = the depth of the deepest leaf
       Ancestor and descendant
            If there is a path from n1 to n2
            n1 is an ancestor of n2, n2 is a descendant of n1
            Proper ancestor and proper descendant
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    Example: UNIX Directory
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         Example: Expression Trees




     Leaves are operands (constants or variables)
     The internal nodes contain operators
     Will not be a binary tree if some operators are not
      binary
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                     Tree Traversal
     Used    to print out the data in a tree in a certain
      order
     Pre-order traversal
         Print the data at the root
         Recursively print out all data in the leftmost subtree
         …
         Recursively print out all data in the rightmost
          subtree
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        Preorder, Postorder and Inorder
      Preorder     traversal
          node, left, right
          prefix expression
               ++a*bc*+*defg
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        Preorder, Postorder and Inorder
      Postorder    traversal    Inorder   traversal
          left, right, node         left, node, right
          postfix expression        infix expression
             abc*+de*f+g*+             a+b*c+d*e+f*g
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     Example: Unix Directory Traversal
          PreOrder           PostOrder
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     Preorder, Postorder and Inorder
              Pseudo Code
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                          Binary Trees
        A tree in which no node can have more than two
         children

                                                            Generic
                                                            binary tree


        The depth of an “average” binary tree is considerably smaller
         than N, even though in the worst case, the depth can be as large
         as N – 1.

                                                            Worst-case
                                                            binary tree
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     Convert a Generic Tree to a Binary Tree
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                        Binary Tree ADT

        Possible operations on the Binary Tree ADT
            Parent, left_child, right_child, sibling, root, etc
        Implementation
           Because a binary tree has at most two children, we can keep
            direct pointers to them
           a linked list is physically a pointer, so is a tree.
        Define a Binary Tree ADT later …
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     A drawing of linked list with one pointer …




     A drawing of binary tree with two pointers …



      Struct BinaryNode {
          double element; // the data
          BinaryNode* left; // left child
          BinaryNode* right; // right child
      }
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            Binary Search Trees (BST)
     A   data structure for efficient searching, inser-
       tion and deletion
      Binary search tree property
           For every node X
           All the keys in its left
            subtree are smaller than
            the key value in X
           All the keys in its right
            subtree are larger than the
            key value in X
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             Binary Search Trees




     A binary search tree   Not a binary search tree
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              Binary Search Trees
         The same set of keys may have different BSTs




      Averagedepth of a node is O(log N)
      Maximum depth of a node is O(N)
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                  Searching BST
      If we are searching for 15, then we are done.
      If we are searching for a key < 15, then we
       should search in the left subtree.
      If we are searching for a key > 15, then we
       should search in the right subtree.
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                  Searching (Find)
      FindX: return a pointer to the node that has
      key X, or NULL if there is no such node



                          find(const double x, BinaryNode* t) const




      Time   complexity: O(height of the tree)
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            Inorder Traversal of BST
      Inorder traversal of BST prints out all the keys
       in sorted order




             Inorder: 2, 3, 4, 6, 7, 9, 13, 15, 17, 18, 20
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                   findMin/ findMax
      Goal: return the node containing the smallest (largest)
       key in the tree
      Algorithm: Start at the root and go left (right) as long as
       there is a left (right) child. The stopping point is the
       smallest (largest) element



                            BinaryNode* findMin(BinaryNode* t) const




        Time complexity = O(height of the tree)
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                             Insertion
      Proceed down the tree as you would with a find
      If X is found, do nothing (or update something)
      Otherwise, insert X at the last spot on the path traversed




        Time complexity = O(height of the tree)
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     void insert(double x, BinaryNode*& t)
     {
         if (t==NULL) t = new BinaryNode(x,NULL,NULL);
         else if (x<t->element) insert(x,t->left);
         else if (t->element<x) insert(x,t->right);
         else ; // do nothing
     }
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                          Deletion
      When we delete a node, we need to consider
      how we take care of the children of the
      deleted node.
          This has to be done such that the property of the
           search tree is maintained.
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      Deletion under Different Cases
      Case   1: the node is a leaf
          Delete it immediately
      Case   2: the node has one child
          Adjust a pointer from the parent to bypass that node
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                        Deletion Case 3
      Case      3: the node has 2 children
           Replace the key of that node with the minimum
            element at the right subtree
           Delete that minimum element
              Has    either no child or only right child because if it has a
                left child, that left child would be smaller and would have
                been chosen. So invoke case 1 or 2.




        Time complexity = O(height of the tree)
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     void remove(double x, BinaryNode*& t)
     {
         if (t==NULL) return;
         if (x<t->element) remove(x,t->left);
         else if (t->element < x) remove (x, t->right);
         else if (t->left != NULL && t->right != NULL) // two children
         {
                t->element = finMin(t->right) ->element;
                remove(t->element,t->right);
         }
         else
         {
                Binarynode* oldNode = t;
                t = (t->left != NULL) ? t->left : t->right;
                delete oldNode;
         }
     }
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     Make a binary or BST ADT …
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                                       For a generic (binary) tree:
     Struct Node {
         double element; // the data
         Node* left; // left child
         Node* right; // right child
     }


     class Tree {
     public:
         Tree();                                        // constructor
         Tree(const Tree& t);
         ~Tree();                                       // destructor


         bool empty() const;


         double root(); // decomposition (access functions)
         Tree& left();                                                         access,
         Tree& right();                                                        selection
         void insert(const double x); // compose x into a tree                    update
         void remove(const double x); // decompose x from a tree
                                            (insert and remove are different from those of BST)
     private:
         Node* root;
     }
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                                       For BST tree:
     Struct Node {
         double element; // the data
         Node* left; // left child
         Node* right; // right child
     }


     class BST {
     public:
         BST();                                       // constructor
         BST(const Tree& t);
         ~BST();                                      // destructor


         bool empty() const;
         double root(); // decomposition (access functions)
         BST left();                                                   access,
         BST right();
                                                                       selection
         bool serch(const double x); // search an element


         void insert(const double x); // compose x into a tree
         void remove(const double x); // decompose x from a tree        update
     private:
         Node* root;
     }                           BST is for efficient search, insertion and removal,
                                 so restricting these functions.
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     class BST {
                                                          Weiss textbook:
     public:
        BST();
        BST(const Tree& t);
        ~BST();


        bool empty() const;
        bool search(const double x); // contains
        void insert(const double x); // compose x into a tree
        void remove(const double x); // decompose x from a tree


     private:
        Struct Node {
                 double element;
                 Node* left;
                 Node* right;


                 Node(…) {…}; // constructuro for Node
        }
        Node* root;


        void insert(const double x, Node*& t) const;     // recursive function
        void remove(…)
        Node* findMin(Node* t);
        void makeEmpty(Node*& t); // recursive ‘destructor’
        bool contains(const double x, Node* t) const;
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        Comments:




        root, left subtree, right subtree are missing:


     1. we can’t write other tree algorithms, is implementation dependent,


 BUT,


     2. this is only for BST (we only need search, insert and remove, may not
     need other tree algorithms)
         so it’s two layers, the public for BST, and the private for Binary Tree.
     3. it might be defined internally in ‘private’ part (actually it’s
     implicitly done).
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     A public non-recursive member function:
      void insert(double x)
      {
          insert(x,root);
      }




     A private recursive member function:
      void insert(double x, BinaryNode*& t)
      {
          if (t==NULL) t = new BinaryNode(x,NULL,NULL);
          else if (x<t->element) insert(x,t->left);
          else if (t->element<x) insert(x,t->right);
          else ; // do nothing
      }

								
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