The binary search tree is a tree in that all the values in the left subtree are less then the value of the root node and values of the right subtree are greater than the value of root node.

binary search tree

as you see in the above example the left subtree has the values lesser then the root node's value and the right subtree has value greater than the root node's value.

Properties of Binary search tree

  • All the keys in the left subtree of the root node are less than the key in the root node.
  • All the keys in the right subtree of the root node are greater than the key in the root node.
  • The Left and right subtrees of the root node are also binary search trees.


Program to implement a binary search tree using the python programming language.


class TreeEmptyError(Exception):
    pass

class Node:
    def __init__(self, value):
        self.info = value
        self.lchild = None        self.rchild = None

class BinarySearchTree:

    def __init__(self):
        self.root = None
    def is_empty(self):
        return self.root is None
    def _insert(self, p, x):
        if p is None:
            p = Node(x)
        elif x < p.info:
            p.lchild = self._insert(p.lchild, x)
        elif x > p.info:
            p.rchild = self._insert(p.rchild, x)
        else:
            print(x, " already present in the tree")
        return p

    def insert1(self, x):
        p = self.root
        par = None        while p is not None:
            par = p
            if x < p.info:
                p = p.lchild
            elif x > p.info:
                p = p.rchild
            else:
                print(x + " already present in the tree")
                return
        temp = Node(x)

        if par is None:
            self.root = temp
        elif x < par.info:
            par.lchild = temp
        else:
            par.rchild = temp

    def search(self, x):
        return self._search(self.root, x) is not None
    def _search(self, p, x):
        if p is None:
            return None        if x < p.info:
            return self._search(p.lchild, x)
        if x > p.info:
            return self._search(p.rchild, x)
        return p

    def search1(self, x):
        p = self.root
        while p is not None:
            if x < p.info:
                p = p.lchild
            elif x > p.info:
                p = p.rchild
            else:
                return True        return False
    def delete(self, x):
        self.root = self._delete(self.root, x)

    def _delete(self, p, x):
        if p is None:
            print(x, " not found")
            return p

        if x < p.info:
            p.lchild = self._delete(p.lchild, x)
        elif x > p.info:
            p.rchild = self._delete(p.rchild, x)
        else:
            if p.lchild is not None and p.rchild is not None:
                s = p.rchild
                while s.lchild is not None:
                    s = s.lchild
                p.info = s.info
                p.rchild = self._delete(p.rchild, s.info)
            else:
                if p.lchild is not None:
                    ch = p.lchild
                else:
                    ch = p.rchild
                p = ch
        return p

    def delete1(self, x):
        p = self.root
        par = None
        while p is not None:
            if x == p.info:
                break            par = p
            if x < p.info:
                p = p.lchild
            else:
                p = p.rchild

            if p is None:
                print(x, " not found")
                return
            if p.lchild is not None and p.rchild is not None:
                ps = p
                s = p.rchild

                while s.lchild is not None:
                    ps = s
                    s = s.lchild
                p.info = s.info
                p = s
                par = ps

            if p.lchild is not None:
                ch = p.lchild
            else:
                ch = p.rchild

            if par is None:
                self.root = ch
            elif p == par.lchild:
                par.lchild = ch
            else:
                par.rchild = ch

    def min1(self):
        if self.is_empty():
            raise TreeEmptyError("Tree is empty")
        p = self.root
        while p.lchild is not None:
            p = p.lchild
        return p.info

    def max1(self):
        if self.is_empty():
            raise TreeEmptyError("Tee is empty")
        p = self.root
        while p.rchild is not None:
            p = p.rchild
        return p.info

    def min2(self):
        if self.is_empty():
            raise TreeEmptyError("Tree is empty")
        return self._min(self.root).info

    def _min(self, p):
        if p.lchild is None:
            return p
        return self._min(p.lchild)

    def max2(self):
        if self.is_empty():
            raise TreeEmptyError("Tree is empty")
        return self._max(self.root).info

    def _max(self, p):
        if p.rchild is None:
            return p
        return self._max(p.rchild)

    def display(self):
        self._display(self.root, 0)
        print()

    def _display(self, p, level):
        if p is None:
            return        self._display(p.rchild, level + 1)
        print()

        for i in range(level):
            print("  ", end='')
        print(p.info)
        self._display(p.lchild, level + 1)

    def preorder(self):
        self._preorder(self.root)
        print()

    def _preorder(self, p):
        if p is None:
            return        print(p.info, " ")
        self._preorder(p.lchild)
        self._preorder(p.rchild)

    def inorder(self):
        self._inorder(self.root)
        print()

    def _inorder(self, p):
        if p is None:
            return        self._inorder(p.lchild)
        print(p.info, " ")
        self._inorder(p.rchild)

    def postorder(self):
        self._postorder(self.root)
        print()

    def _postorder(self, p):
        if p is None:
            return        self._postorder(p.lchild)
        self._postorder(p.rchild)
        print(p.info, " ")

    def height(self):
        return self._height(self.root)

    def _height(self, p):
        if p is None:
            return 0
        hL = self._height(p.lchild)
        hR = self._height(p.rchild)

        if hL > hR:
            return 1 + hL
        else:
            return 1 + hR


###################################
bst = BinarySearchTree()

while True:
    print("1.Display Tree")
    print("2.Search(Iterative)")
    print("3.Search(Recursive)")
    print("4.Insert a new node(Iterative)")
    print("5.Insert a noew node(Recursive)")
    print("6.Delete a node(Iterative)")
    print("7.Delete a node(Recursive)")
    print("8.Find Minimum key(Iterative)")
    print("9.Find Minimum key(Recursive)")
    print("10.Find Maximum key(Iterative)")
    print("11.Find Maximum key(Recursive)")
    print("12.Preorder Traversal")
    print("13.Inorder Traversal")
    print("14.Postoder Traversal")
    print("15.Height of tree")
    print("16.Quit")
    choice = int(input("Enter your choice : "))

    if choice == 1:
        bst.display()
    elif choice == 2:
        x = int(input("Enter the key to be searched : "))
        if bst.search1(x):
            print("Key found")
        else:
            print("Key not found")
    elif choice == 3:
        x = int(input("Enter the key to be searched : "))
        if bst.search(x):
            print("Key found")
        else:
            print("Key not found")

    elif choice == 4:
        x = int(input("Etner the key to be inserte : "))
        bst.insert1(x)
    elif choice == 5:
        x = int(input("Enter the key to be inserted : "))
        bst.insert1(x)
    elif choice == 6:
        x = int(input("Enter the element to be deleted : "))
        bst.delete1(x)
    elif choice == 7:
        x = int(input("Enter the element to be deleted : "))
        bst.delete(x)
    elif choice == 8:
        print("Minimum key is ", bst.min1())
    elif choice == 9:
        print("Minimum key is ", bst.min2())
    elif choice == 10:
        print("Maximum key is ", bst.max1())
    elif choice == 11:
        print("Maximum key is ", bst.max2())
    elif choice == 12:
        bst.preorder()
    elif choice == 13:
        bst.inorder()
    elif choice == 14:
        bst.postorder()
    elif choice == 15:
        print("Height of tree is ", bst.height())
    elif choice == 16:
        break    else:
        print("wrong choice")
    print()