Kth Smallest Element in a BST

Given a binary search tree, write a function kthSmallest to find the kth smallest element in it.

Note:
You may assume k is always valid, 1 ≤ k ≤ BST’s total elements.

Follow up:
What if the BST is modified (insert/delete operations) often and you need to find the kth smallest frequently? How would you optimize the kthSmallest routine?

Hint:

1. Try to utilize the property of a BST.
2. What if you could modify the BST node’s structure?
3. The optimal runtime complexity is O(height of BST).

Credits:
Special thanks to @ts for adding this problem and creating all test cases.

Naive approach, convert BST to a sequence, find the Kth element.

O(n) in space, O(n) in time.

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     TreeNode *left;
 *     TreeNode *right;
 *     TreeNode(int x) : val(x), left(NULL), right(NULL) {}
 * };
 */
class Solution {
public:
    int kthSmallest(TreeNode* root, int k) {
        vector<TreeNode *> seq;
        convert2vector(root, seq);
        return seq[k - 1]->val;
    }
    void convert2vector(TreeNode * root, vector<TreeNode *>& seq){
        if(root == nullptr){
            return ;
        }
        convert2vector(root->left, seq);
        seq.push_back(root);
        convert2vector(root->right, seq);
    }
};

A better approach. Calculate the number of nodes in subtree of each node. Then use binary search.

O(h) in time for each search. h is the height of the tree. O(n) in generating the number of nodes information.