Find kth smallest element in a binary search tree in Optimum way

I need to find the kth smallest element in the binary search tree without using any static/global variable. How to achieve it efficiently? The solution that I have in my mind is doing the operation in O(n), the worst case since I am planning to do an inorder traversal of the entire tree. But deep down I feel that I am not using the BST property here. Is my assumptive solution correct or is there a better one available ?

Here's just an outline of the idea:

In a BST, the left subtree of node T contains only elements smaller than the value stored in T. If k is smaller than the number of elements in the left subtree, the kth smallest element must belong to the left subtree. Otherwise, if k is larger, then the kth smallest element is in the right subtree.

We can augment the BST to have each node in it store the number of elements in its left subtree (assume that the left subtree of a given node includes that node). With this piece of information, it is simple to traverse the tree by repeatedly asking for the number of elements in the left subtree, to decide whether to do recurse into the left or right subtree.

Now, suppose we are at node T:

1. If k == num_elements(left subtree of T), then the answer we're looking for is the value in node T.
2. If k > num_elements(left subtree of T), then obviously we can ignore the left subtree, because those elements will also be smaller than the kth smallest. So, we reduce the problem to finding the k - num_elements(left subtree of T) smallest element of the right subtree.
3. If k < num_elements(left subtree of T), then the kth smallest is somewhere in the left subtree, so we reduce the problem to finding the kth smallest element in the left subtree.

Complexity analysis:

This takes O(depth of node) time, which is O(log n) in the worst case on a balanced BST, or O(log n) on average for a random BST.

A BST requires O(n) storage, and it takes another O(n) to store the information about the number of elements. All BST operations take O(depth of node) time, and it takes O(depth of node) extra time to maintain the "number of elements" information for insertion, deletion or rotation of nodes. Therefore, storing information about the number of elements in the left subtree keeps the space and time complexity of a BST.

A simpler solution would be to do an inorder traversal and keep track of the element currently to be printed (without printing it). When we reach k, print the element and skip rest of tree traversal.

void findK(Node* p, int* k) {
  if(!p || k < 0) return;
  findK(p->left, k);
  --k;
  if(k == 0) { 
    print p->data;
    return;  
  } 
  findK(p->right, k); 
}

public int ReturnKthSmallestElement1(int k)
    {
        Node node = Root;

        int count = k;

        int sizeOfLeftSubtree = 0;

        while(node != null)
        {

            sizeOfLeftSubtree = node.SizeOfLeftSubtree();

            if (sizeOfLeftSubtree + 1 == count)
                return node.Value;
            else if (sizeOfLeftSubtree < count)
            {
                node = node.Right;
                count -= sizeOfLeftSubtree+1;
            }
            else
            {
                node = node.Left;
            }
        }
            
        return -1;
    }

this is my implementation in C# based on the algorithm above just thought I'd post it so people can understand better it works for me

thank you IVlad

//add a java version without recursion

public static <T> void find(TreeNode<T> node, int num){
	Stack<TreeNode<T>> stack = new Stack<TreeNode<T>>();
	
	TreeNode<T> current = node;
	int tmp = num;
	
	while(stack.size() > 0 || current!=null){
		if(current!= null){
			stack.add(current);
			current = current.getLeft();
		}else{
			current = stack.pop();
			tmp--;
			
			if(tmp == 0){
				System.out.println(current.getValue());
				return;
			}
			
			current = current.getRight();
		}
	}
}

A simpler solution would be to do an inorder traversal and keep track of the element currently to be printed with a counter k. When we reach k, print the element. The runtime is O(n). Remember the function return type can not be void, it has to return its updated value of k after each recursive call. A better solution to this would be an augmented BST with a sorted position value at each node.

public static int kthSmallest (Node pivot, int k){
    if(pivot == null )
        return k;   
    k = kthSmallest(pivot.left, k);
    k--;
    if(k == 0){
        System.out.println(pivot.value);
    }
    k = kthSmallest(pivot.right, k);
    return k;
}

You can use iterative inorder traversal: http://en.wikipedia.org/wiki/Tree_traversal#Iterative_Traversal with a simple check for kth element after poping a node out of the stack.

Given just a plain binary search tree, about all you can do is start from the smallest, and traverse upward to find the right node.

If you're going to do this very often, you can add an attribute to each node signifying how many nodes are in its left sub-tree. Using that, you can descend the tree directly to the correct node.

Recursive In-order Walk with a counter

Time Complexity: O( N ), N is the number of nodes
Space Complexity: O( 1 ), excluding the function call stack

The idea is similar to @prasadvk solution, but it has some shortcomings (see notes below), so I am posting this as a separate answer.

// Private Helper Macro
#define testAndReturn( k, counter, result )                         \
    do { if( (counter == k) && (result == -1) ) {                   \
        result = pn->key_;                                          \
        return;                                                     \
    } } while( 0 )

// Private Helper Function
static void findKthSmallest(
    BstNode const * pn, int const k, int & counter, int & result ) {

    if( ! pn ) return;

    findKthSmallest( pn->left_, k, counter, result );
    testAndReturn( k, counter, result );

    counter += 1;
    testAndReturn( k, counter, result );

    findKthSmallest( pn->right_, k, counter, result );
    testAndReturn( k, counter, result );
}

// Public API function
void findKthSmallest( Bst const * pt, int const k ) {
    int counter = 0;
    int result = -1;        // -1 := not found
    findKthSmallest( pt->root_, k, counter, result );
    printf("%d-th element: element = %d\n", k, result );
}

Notes (and differences from @prasadvk's solution):

1. if( counter == k ) test is required at three places: (a) after left-subtree, (b) after root, and (c) after right subtree. This is to ensure that kth element is detected for all locations, i.e. irrespective of the subtree it is located.

2. if( result == -1 ) test required to ensure only the result element is printed, otherwise all the elements starting from the kth smallest up to the root are printed.

For not balanced searching tree, it takes O(n).

For balanced searching tree, it takes O(k + log n) in the worst case but just O(k) in Amortized sense.

Having and managing the extra integer for every node: the size of the sub-tree gives O(log n) time complexity. Such balanced searching tree is usually called RankTree.

In general, there are solutions (based not on tree).

Regards.

This works well: status : is the array which holds whether element is found. k : is kth element to be found. count : keeps track of number of nodes traversed during the tree traversal.

int kth(struct tree* node, int* status, int k, int count)
{
    if (!node) return count;
    count = kth(node->lft, status, k, count);  
    if( status[1] ) return status[0];
    if (count == k) { 
        status[0] = node->val;
        status[1] = 1;
        return status[0];
    }
    count = kth(node->rgt, status, k, count+1);
    if( status[1] ) return status[0];
    return count;
}

Well here is my 2 cents...

int numBSTnodes(const Node* pNode){
     if(pNode == NULL) return 0;
     return (numBSTnodes(pNode->left)+numBSTnodes(pNode->right)+1);
}


//This function will find Kth smallest element
Node* findKthSmallestBSTelement(Node* root, int k){
     Node* pTrav = root;
     while(k > 0){
         int numNodes = numBSTnodes(pTrav->left);
         if(numNodes >= k){
              pTrav = pTrav->left;
         }
         else{
              //subtract left tree nodes and root count from 'k'
              k -= (numBSTnodes(pTrav->left) + 1);
              if(k == 0) return pTrav;
              pTrav = pTrav->right;
        }
        
        return NULL;
 }

While this is definitely not the optimal solution to the problem, it is another potential solution which I thought some people might find interesting:

/**
 * Treat the bst as a sorted list in descending order and find the element 
 * in position k.
 *
 * Time complexity BigO ( n^2 )
 *
 * 2n + sum( 1 * n/2 + 2 * n/4 + ... ( 2^n-1) * n/n ) = 
 * 2n + sigma a=1 to n ( (2^(a-1)) * n / 2^a ) = 2n + n(n-1)/4
 *
 * @param t The root of the binary search tree.
 * @param k The position of the element to find.
 * @return The value of the element at position k.
 */
public static int kElement2( Node t, int k ) {
    int treeSize = sizeOfTree( t );

    return kElement2( t, k, treeSize, 0 ).intValue();
}

/**
 * Find the value at position k in the bst by doing an in-order traversal 
 * of the tree and mapping the ascending order index to the descending order 
 * index.
 *
 *
 * @param t Root of the bst to search in.
 * @param k Index of the element being searched for.
 * @param treeSize Size of the entire bst.
 * @param count The number of node already visited.
 * @return Either the value of the kth node, or Double.POSITIVE_INFINITY if 
 *         not found in this sub-tree.
 */
private static Double kElement2( Node t, int k, int treeSize, int count ) {
    // Double.POSITIVE_INFINITY is a marker value indicating that the kth 
    // element wasn't found in this sub-tree.
    if ( t == null )
        return Double.POSITIVE_INFINITY;

    Double kea = kElement2( t.getLeftSon(), k, treeSize, count );

    if ( kea != Double.POSITIVE_INFINITY )
        return kea;

    // The index of the current node.
    count += 1 + sizeOfTree( t.getLeftSon() );

    // Given any index from the ascending in order traversal of the bst, 
    // treeSize + 1 - index gives the
    // corresponding index in the descending order list.
    if ( ( treeSize + 1 - count ) == k )
        return (double)t.getNumber();

    return kElement2( t.getRightSon(), k, treeSize, count );
}

signature:

Node * find(Node* tree, int *n, int k);

call as:

*n = 0;
kthNode = find(root, n, k);

definition:

Node * find ( Node * tree, int *n, int k)
{
   Node *temp = NULL;

   if (tree->left && *n<k)
      temp = find(tree->left, n, k);

   *n++;

   if(*n==k)
      temp = root;

   if (tree->right && *n<k)
      temp = find(tree->right, n, k);

   return temp;
}

This is what I though and it works. It will run in o(log n )

public static int FindkThSmallestElemet(Node root, int k)
    {
        int count = 0;
        Node current = root;

        while (current != null)
        {
            count++;
            current = current.left;
        }
        current = root;

        while (current != null)
        {
            if (count == k)
                return current.data;
            else
            {
                current = current.left;
                count--;
            }
        }

        return -1;


    } // end of function FindkThSmallestElemet

Well we can simply use the in order traversal and push the visited element onto a stack. pop k number of times, to get the answer.

we can also stop after k elements

Solution for complete BST case :-

Node kSmallest(Node root, int k) {
  int i = root.size(); // 2^height - 1, single node is height = 1;
  Node result = root;
  while (i - 1 > k) {
    i = (i-1)/2;  // size of left subtree
    if (k < i) {
      result = result.left;
    } else {
      result = result.right;
      k -= i;
    }  
  }
  return i-1==k ? result: null;
}

The Linux Kernel has an excellent augmented red-black tree data structure that supports rank-based operations in O(log n) in linux/lib/rbtree.c.

A very crude Java port can also be found at http://code.google.com/p/refolding/source/browse/trunk/core/src/main/java/it/unibo/refolding/alg/RbTree.java, together with RbRoot.java and RbNode.java. The n'th element can be obtained by calling RbNode.nth(RbNode node, int n), passing in the root of the tree.

Here's a concise version in C# that returns the k-th smallest element, but requires passing k in as a ref argument (it's the same approach as @prasadvk):

Node FindSmall(Node root, ref int k)
{
    if (root == null || k < 1)
        return null;

    Node node = FindSmall(root.LeftChild, ref k);
    if (node != null)
        return node;

    if (--k == 0)
        return node ?? root;
    return FindSmall(root.RightChild, ref k);
}

It's O(log n) to find the smallest node, and then O(k) to traverse to k-th node, so it's O(k + log n).

http://www.geeksforgeeks.org/archives/10379

this is the exact answer to this question:-

1.using inorder traversal on O(n) time 2.using Augmented tree in k+log n time

I couldn't find a better algorithm..so decided to write one :) Correct me if this is wrong.

class KthLargestBST{
protected static int findKthSmallest(BSTNode root,int k){//user calls this function
	int [] result=findKthSmallest(root,k,0);//I call another function inside
	return result[1];
}
private static int[] findKthSmallest(BSTNode root,int k,int count){//returns result[]2 array containing count in rval[0] and desired element in rval[1] position.
	if(root==null){
		int[]  i=new int[2];
		i[0]=-1;
		i[1]=-1;
		return i;
	}else{
		int rval[]=new int[2];
		int temp[]=new int[2];
		rval=findKthSmallest(root.leftChild,k,count);
		if(rval[0]!=-1){
			count=rval[0];
		}
		count++;
		if(count==k){
			rval[1]=root.data;
		}
		temp=findKthSmallest(root.rightChild,k,(count));
		if(temp[0]!=-1){
			count=temp[0];
		}
		if(temp[1]!=-1){
			rval[1]=temp[1];
		}
		rval[0]=count;
		return rval;
	}
}
public static void main(String args[]){
	BinarySearchTree bst=new BinarySearchTree();
	bst.insert(6);
	bst.insert(8);
	bst.insert(7);
	bst.insert(4);
	bst.insert(3);
	bst.insert(4);
	bst.insert(1);
	bst.insert(12);
	bst.insert(18);
	bst.insert(15);
	bst.insert(16);
	bst.inOrderTraversal();
	System.out.println();
	System.out.println(findKthSmallest(bst.root,11));
}

}

Here is the java code,

max(Node root, int k) - to find kth largest

min(Node root, int k) - to find kth Smallest

static int count(Node root){
    if(root == null)
        return 0;
    else
        return count(root.left) + count(root.right) +1;
}
static int max(Node root, int k) {
    if(root == null)
        return -1;
    int right= count(root.right);

    if(k == right+1)
        return root.data;
    else if(right < k)
        return max(root.left, k-right-1);
    else return max(root.right, k);
}

static int min(Node root, int k) {
    if (root==null)
        return -1;

    int left= count(root.left);
    if(k == left+1)
        return root.data;
    else if (left < k)
        return min(root.right, k-left-1);
    else
        return min(root.left, k);
}

this would work too. just call the function with maxNode in the tree

def k_largest(self, node , k): if k < 0 : return None
if k == 0: return node else: k -=1 return self.k_largest(self.predecessor(node), k)

I think this is better than the accepted answer because it doesn't need to modify the original tree node to store the number of it's children nodes.

We just need to use the in-order traversal to count the smallest node from the left to right, stop searching once the count equals to K.

private static int count = 0;
public static void printKthSmallestNode(Node node, int k){
    if(node == null){
        return;
    }

    if( node.getLeftNode() != null ){
        printKthSmallestNode(node.getLeftNode(), k);
    }

    count ++ ;
    if(count <= k )
        System.out.println(node.getValue() + ", count=" + count + ", k=" + k);

    if(count < k  && node.getRightNode() != null)
        printKthSmallestNode(node.getRightNode(), k);
}

Best approach is already there.But I'd like to add a simple Code for that

int kthsmallest(treenode *q,int k){
int n = size(q->left) + 1;
if(n==k){
    return q->val;
}
if(n > k){
    return kthsmallest(q->left,k);
}
if(n < k){
    return kthsmallest(q->right,k - n);
}

}

int size(treenode *q){
if(q==NULL){
    return 0;
}
else{
    return ( size(q->left) + size(q->right) + 1 );
}}

Using auxiliary Result class to track if node is found and current k.

public class KthSmallestElementWithAux {

public int kthsmallest(TreeNode a, int k) {
    TreeNode ans = kthsmallestRec(a, k).node;
    if (ans != null) {
        return ans.val;
    } else {
        return -1;
    }
}

private Result kthsmallestRec(TreeNode a, int k) {
    //Leaf node, do nothing and return
    if (a == null) {
        return new Result(k, null);
    }

    //Search left first
    Result leftSearch = kthsmallestRec(a.left, k);

    //We are done, no need to check right.
    if (leftSearch.node != null) {
        return leftSearch;
    }

    //Consider number of nodes found to the left
    k = leftSearch.k;

    //Check if current root is the solution before going right
    k--;
    if (k == 0) {
        return new Result(k - 1, a);
    }

    //Check right
    Result rightBalanced = kthsmallestRec(a.right, k);

    //Consider all nodes found to the right
    k = rightBalanced.k;

    if (rightBalanced.node != null) {
        return rightBalanced;
    }

    //No node found, recursion will continue at the higher level
    return new Result(k, null);

}

private class Result {
    private final int k;
    private final TreeNode node;

    Result(int max, TreeNode node) {
        this.k = max;
        this.node = node;
    }
}
}

Python Solution Time Complexity : O(n) Space Complexity : O(1)

Idea is to use Morris Inorder Traversal

class Solution(object):
def inorderTraversal(self, current , k ):
    while(current is not None):    #This Means we have reached Right Most Node i.e end of LDR traversal
        
        if(current.left is not None):  #If Left Exists traverse Left First
            pre = current.left   #Goal is to find the node which will be just before the current node i.e predecessor of current node, let's say current is D in LDR goal is to find L here
            while(pre.right is not None and pre.right != current ): #Find predecesor here
                pre = pre.right
            if(pre.right is None):  #In this case predecessor is found , now link this predecessor to current so that there is a path and current is not lost
                pre.right = current
                current = current.left
            else:                   #This means we have traverse all nodes left to current so in LDR traversal of L is done
                k -= 1
                if(k == 0):
                    return current.val
                pre.right = None       #Remove the link tree restored to original here 
                current = current.right
        else:               #In LDR  LD traversal is done move to R 
            k -= 1
            if(k == 0):
                return current.val
            current = current.right
                
    return 0

def kthSmallest(self, root, k):
    return self.inorderTraversal( root , k  )

public int kthSmallest(TreeNode root, int k) {
     
    LinkedList<TreeNode> stack = new LinkedList<TreeNode>();

    while (true) {
      while (root != null) {
        stack.push(root);
        root = root.left;
      }
      root = stack.pop();
      k = k - 1;
      if (k == 0) return root.val;
      root = root.right;
    }

}     

Here are the steps:

1.Add a field to each node indicating the size of the tree it roots. This supports operation in O(logN) average time.

2.To save space, one field indicating the size of a node it roots is enough. We don't need to save both the left subtree and right subtree size.

3.Do an inorder traversal until LeftTree == K, LeftTree = Size(T->Left) + 1.

4.Here is the sample code:

int Size(SearchTree T)
{
    if(T == NULL) return 0;
    return T->Size;
}
Position KthSmallest(SearchTree T, int K)
{
    if(T == NULL) return NULL;

    int LeftTree;
    LeftTree = Size(T->Left) + 1;

    if(LeftTree == K) return T;

    if(LeftTree > K){ 
        T = KthSmallest(T->Left, K); 
    }else if(LeftTree < K){ 
        T = KthSmallest(T->Right, K - LeftTree);
    }   

    return T;
}

5.Similarly, we can also get the KthLargest function.

i wrote a neat function to calculate the kth smallest element. I uses in-order traversal and stops when the it reaches the kth smallest element.

void btree::kthSmallest(node* temp, int& k){
if( temp!= NULL)   {
 kthSmallest(temp->left,k);       
 if(k >0)
 {
     if(k==1)
    {
      cout<<temp->value<<endl;
      return;
    }

    k--;
 }
 
 kthSmallest(temp->right,k);  }}

public static Node kth(Node n, int k){
    Stack<Node> s=new Stack<Node>();
    int countPopped=0;
    while(!s.isEmpty()||n!=null){
      if(n!=null){
        s.push(n);
        n=n.left;
      }else{
        node=s.pop();
        countPopped++;
        if(countPopped==k){
            return node;
        }
        node=node.right;
       
      }
  }

}

 public int printInorder(Node node, int k) 
    { 
        if (node == null || k <= 0) //Stop traversing once you found the k-th smallest element
            return k; 

        /* first recur on left child */
        k = printInorder(node.left, k); 

        k--;
        if(k == 0) {  
            System.out.print(node.key);
        }

        /* now recur on right child */
        return printInorder(node.right, k);
    } 

This java recursive algorithm, stops the recursion once the k-th smallest element is found.

int RecPrintKSmallest(Node_ptr head,int k){
  if(head!=NULL){
	k=RecPrintKSmallest(head->left,k);
	if(k>0){
      printf("%c ",head->Node_key.key);
      k--;
	}
	k=RecPrintKSmallest(head->right,k);
  }
  return k;
}

public TreeNode findKthElement(TreeNode root, int k){
    if((k==numberElement(root.left)+1)){
        return root;
    }
    else if(k>numberElement(root.left)+1){
        findKthElement(root.right,k-numberElement(root.left)-1);
    }
    else{
        findKthElement(root.left, k);
    }
}

public int numberElement(TreeNode node){
    if(node==null){
        return 0;
    }
    else{
        return numberElement(node.left) + numberElement(node.right) + 1;
    }
}

For a binary search tree, an inorder traversal will return elements ... in order.

Just do an inorder traversal and stop after traversing k elements.

O(1) for constant values of k.