<p>Given an integer <code>n</code>, return <em>all the structurally unique <strong>BST&#39;</strong>s (binary search trees), which has exactly </em><code>n</code><em> nodes of unique values from</em> <code>1</code> <em>to</em> <code>n</code>. Return the answer in <strong>any order</strong>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/01/18/uniquebstn3.jpg" style="width: 600px; height: 148px;" />
<pre>
<strong>Input:</strong> n = 3
<strong>Output:</strong> [[1,null,2,null,3],[1,null,3,2],[2,1,3],[3,1,null,null,2],[3,2,null,1]]
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 1
<strong>Output:</strong> [[1]]
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 8</code></li>
</ul>

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     TreeNode *left;
 *     TreeNode *right;
 *     TreeNode() : val(0), left(nullptr), right(nullptr) {}
 *     TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}
 *     TreeNode(int x, TreeNode *left, TreeNode *right) : val(x), left(left), right(right) {}
 * };
 */
class Solution {
public:
    vector<TreeNode*> generateTrees(int n) {
        if (n == 0) {
            return vector<TreeNode*>();
        }
        
        unordered_map<string, vector<TreeNode*>> memo;

        return generateTreesHelper(1, n, memo);        
    }

private:
    vector<TreeNode*> generateTreesHelper(int start, int end, unordered_map<string, vector<TreeNode*>>& memo) {
        string key = to_string(start) + "-" + to_string(end);
        if (memo.find(key) != memo.end()) {
            return memo[key];
        }
        
        vector<TreeNode*> trees;
        if (start > end) {
            trees.push_back(nullptr);
            return trees;
        }
        
        for (int rootVal = start; rootVal <= end; rootVal++) {
            vector<TreeNode*> leftTrees = generateTreesHelper(start, rootVal - 1, memo);
            vector<TreeNode*> rightTrees = generateTreesHelper(rootVal + 1, end, memo);
            
            for (TreeNode* leftTree : leftTrees) {
                for (TreeNode* rightTree : rightTrees) {
                    TreeNode* root = new TreeNode(rootVal);
                    root->left = leftTree;
                    root->right = rightTree;
                    trees.push_back(root);
                }
            }
        }
        
        memo[key] = trees;
        return trees;
    }    
};
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