Unique Binary Search Trees II

This approach uses a straightforward recursive method to construct all possible Binary Search Trees. The core idea is based on the definition of a BST: for any chosen root i from the range [1, n], all values smaller than i must go into the left subtree, and all values larger than i must go into the right subtree. We can recursively generate all possible left and right subtrees and then combine them for each choice of root i.

1. Define a recursive function, say generate(start, end), that returns a list of all unique BSTs for values in the range [start, end].
2. The main function calls generate(1, n).
3. In generate(start, end):
   • If start > end, it signifies an empty subtree. Return a list containing a single null element to represent this.
   • Initialize an empty list, all_trees, to store the generated BSTs for the current range.
   • Iterate with a loop variable i from start to end. This i will serve as the root of a BST.
   • Make a recursive call to generate(start, i-1) to get all possible left subtrees.
   • Make another recursive call to generate(i+1, end) to get all possible right subtrees.
   • Nest two loops to iterate through each left_tree from the list of left subtrees and each right_tree from the list of right subtrees.
   • Inside the loops, for each pair of (left_tree, right_tree), create a new TreeNode with value i. Set its left child to left_tree and its right child to right_tree.
   • Add this newly constructed tree to the all_trees list.
4. After the loops complete, return the all_trees list.

The algorithm defines a helper function, generateSubtrees(start, end), which is responsible for generating all BSTs using numbers from start to end. For each number i in this range, we consider it as the root. Then, we recursively call the function to generate all possible left subtrees from the range [start, i-1] and all possible right subtrees from [i+1, end]. Finally, we combine each left subtree with each right subtree to form a new tree rooted at i.

The base case for the recursion is when start > end, which represents an empty range. In this case, we return a list containing null to signify an empty subtree. This null is crucial for the loops that combine subtrees, ensuring they execute correctly even when one side is empty.

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     int val;
 *     TreeNode left;
 *     TreeNode right;
 *     TreeNode() {}
 *     TreeNode(int val) { this.val = val; }
 *     TreeNode(int val, TreeNode left, TreeNode right) {
 *         this.val = val;
 *         this.left = left;
 *         this.right = right;
 *     }
 * }
 */
class Solution {
    public List<TreeNode> generateTrees(int n) {
        if (n == 0) {
            return new ArrayList<>();
        }
        return generateSubtrees(1, n);
    }

    private List<TreeNode> generateSubtrees(int start, int end) {
        List<TreeNode> allTrees = new ArrayList<>();
        if (start > end) {
            allTrees.add(null);
            return allTrees;
        }

        // Pick each number in the range as a root
        for (int i = start; i <= end; i++) {
            // Generate all possible left subtrees
            List<TreeNode> leftSubtrees = generateSubtrees(start, i - 1);
            // Generate all possible right subtrees
            List<TreeNode> rightSubtrees = generateSubtrees(i + 1, end);

            // Combine each left subtree with each right subtree
            for (TreeNode left : leftSubtrees) {
                for (TreeNode right : rightSubtrees) {
                    TreeNode root = new TreeNode(i);
                    root.left = left;
                    root.right = right;
                    allTrees.add(root);
                }
            }
        }
        return allTrees;
    }
}