<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)
    {
        this->cache = vector(n + 1, vector<vector<TreeNode*>*>(n + 1, nullptr));
        return this->helper(1, n);
    }

private:
    vector<TreeNode *> helper(int start, int end)
    {
        vector<TreeNode *> *res = new vector<TreeNode *>;
        if (start > end)
        {
            res->push_back(nullptr);
            return *res;
        }
        if (this->cache[start][end] != nullptr) {
            return *(this->cache[start][end]);
        }
        for (int i = start; i <= end; i++)
        {
            vector<TreeNode *> left = this->helper(start, i - 1);
            vector<TreeNode *> right = this->helper(i + 1, end);
            for (TreeNode* l : left) {
                for (TreeNode* r : right) {
                    TreeNode *cur = new TreeNode(i);
                    cur->left = l;
                    cur->right = r;
                    res->push_back(cur);
                }
            }
        }
        this->cache[start][end] = res;
        return *res;
    }
    vector<vector<vector<TreeNode*>*>> cache;
};
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