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This approach uses recursion to build all possible full binary trees for a given number of nodes. Since the problem exhibits overlapping subproblems, we can optimize the solution using memoization to store the results of previously solved subproblems. This significantly reduces the number of calculations needed.
The time complexity is O(2^N) due to the recursive generation of all possible subtrees. However, memoization helps to reduce the number of redundant calculations. The space complexity is also O(2^N) due to the storage of previously computed subtrees in the memo dictionary.
1class TreeNode {
2 constructor(val = 0, left = null, right = null) {
3 this.val = val;
4 this.left = left;
5 this.right = right;
6 }
7}
8
9const memo = {};
10
11function allPossibleFBT(N) {
12 if (N % 2 === 0) return [];
13 if (N === 1) return [new TreeNode(0)];
14 if (memo[N]) return memo[N];
15
16 const result = [];
17 for (let left = 1; left < N; left += 2) {
18 for (let leftNode of allPossibleFBT(left)) {
19 for (let rightNode of allPossibleFBT(N - 1 - left)) {
20 const root = new TreeNode(0);
21 root.left = leftNode;
22 root.right = rightNode;
23 result.push(root);
24 }
25 }
26 }
27 memo[N] = result;
28 return result;
29}The JavaScript function allPossibleFBT implements a similar memoized recursive approach as the Python version. It creates instances of TreeNode for the roots of subtrees, and memoizes results to avoid redundant calculations for the same N. This helps in creating and efficiently storing valid trees.
This approach uses dynamic programming through an iterative method to build trees from the bottom up. We generate trees with increasingly larger numbers of nodes by combining smaller full binary trees, ensuring that each construction step builds on the last.
This method maintains a time complexity of O(2^N) and a space complexity of O(2^N) due to stored references of constructed trees. While iterative, it effectively mirrors the recursive/memoized strategies through explicit storage of previous calculations.
1import java.util.*;
2
In this Java solution, an array of lists dp is used to store trees. Each tree is represented by a TreeNode with zero value. This iterative process combines smaller trees at each step to build bigger trees, ensuring that all trees have the characteristic of full binary trees, which inherently coordinates the nodes' distribution between left and right subtrees.