Problem Description
Given an integer n, return a list of all possible full binary trees with n nodes. Each node of each tree in the answer must have Node.val == 0. Each element of the answer is the root node of one possible tree. You may return the final list of trees in any order. A full binary tree is a binary tree where each node has exactly 0 or 2 children.
Key Insights
- A full binary tree must have an odd number of nodes (1, 3, 5, ...).
- Each full binary tree can be constructed by choosing a left subtree and a right subtree, both of which are also full binary trees.
- The recursive nature of the problem allows for the generation of trees by combining smaller full binary trees.
- Memoization can be utilized to store previously computed trees for certain node counts to avoid redundant calculations.
Space and Time Complexity
Time Complexity: O(2^n) - The number of full binary trees grows exponentially with n, as each node can lead to multiple combinations of left and right subtrees. Space Complexity: O(n) - Space is required for storing the trees and the recursion stack.
Solution
To solve the problem, we can use a recursive approach. We define a function that generates all possible full binary trees for a given number of nodes. The function iterates through possible sizes for the left subtree and computes the size of the right subtree accordingly. For each combination of left and right subtrees, a new tree is created with a root node. This process continues recursively until the base case is reached (1 node).
The data structures used include:
- A list to store the generated full binary trees.
- A TreeNode class to represent each node in the binary tree.