#894

Medium

auto_awesomeBinary-tree traversal and state tracking

LeetCode Problem Workspace

All Possible Full Binary Trees

Generate all possible full binary trees with n nodes, focusing on dynamic programming and binary tree traversal.

dynamic programming tree recursion memoization binary tree

Problem Statement

You are given an integer n and asked to return a list of all possible full binary trees with n nodes. A full binary tree is one where every node has either 0 or 2 children. Each tree's root node must have a value of 0. The final output should contain all unique full binary trees, and the list can be returned in any order.

Your goal is to construct all possible valid full binary trees with exactly n nodes. Each element in the answer represents the root node of one possible tree. Constraints limit n to the range of 1 to 20, meaning the solution must be efficient enough to handle these inputs.

Examples

Example 1

Input: n = 7

Output: [[0,0,0,null,null,0,0,null,null,0,0],[0,0,0,null,null,0,0,0,0],[0,0,0,0,0,0,0],[0,0,0,0,0,null,null,null,null,0,0],[0,0,0,0,0,null,null,0,0]]

Example details omitted.

Example 2

Input: n = 3

Output: [[0,0,0]]

Example details omitted.

Constraints

1 <= n <= 20

Solution Approach

Recursive Tree Construction

The problem can be approached recursively by considering the number of nodes in the left and right subtrees for each possible root node. Each subtree must also be a full binary tree. We recursively compute all combinations of left and right subtrees for each node, ensuring that the total number of nodes adds up to n.

Dynamic Programming with Memoization

Given the overlapping subproblems in this problem, memoization is used to store the results of previously computed subtrees. This prevents redundant calculations by reusing previously computed results for the same subtree sizes, leading to an optimized solution.

Combination of Subtrees

For each n, generate all valid left and right subtrees by considering all possible splits of n-1 nodes (since one node is the root). The total number of possible trees is determined by combining all possible left subtrees with all possible right subtrees. The resulting trees are stored for reuse during the computation of larger trees.

Complexity Analysis

Metric	Value
Time	O(2^{n/2})
Space	O(n \cdot 2^{n/2})

The time complexity is O(2^{n/2}) because each recursive call generates all combinations of left and right subtrees, leading to exponential growth. The space complexity is O(n * 2^{n/2}) due to the storage required for memoized results and the tree structure itself.

What Interviewers Usually Probe

Understand recursion and dynamic programming to optimize tree construction.
Recognize overlapping subproblems and apply memoization effectively.
Be able to explain how combining subtrees leads to generating full binary trees.

Common Pitfalls or Variants

Common pitfalls

Forgetting to memoize the results of subproblems, leading to unnecessary recalculations.
Incorrectly handling the base case or subtree splitting, resulting in invalid trees.
Overlooking the efficiency trade-off, which could result in performance issues for larger values of n.

Follow-up variants

Modify the problem to return all possible unique full binary trees of height h instead of a fixed number of nodes.
Alter the problem by adding constraints on the values of the nodes (not just 0).
Change the tree structure to allow a binary tree with nodes having 1, 2, or more children.

FAQ

What is a full binary tree?

A full binary tree is a binary tree where each node has either 0 or 2 children, with no nodes having only one child.

How can I optimize my solution for All Possible Full Binary Trees?

You can optimize the solution using dynamic programming and memoization, which reduces redundant calculations by storing previously computed subtrees.

What is the key idea behind solving this problem?

The key idea is recursively constructing valid subtrees and combining them efficiently, leveraging memoization to optimize repeated calculations.

How do I handle large inputs for this problem?

To handle larger values of n, it's crucial to implement an optimized approach using dynamic programming with memoization, which reduces the time complexity significantly.

What pattern does this problem involve?

This problem is a classic example of binary tree traversal and state tracking, where dynamic programming and recursion are essential for an efficient solution.

terminal

Solution

Solution 1: Memoization Search

If $n=1$, return a list with a single node directly.

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# Definition for a binary tree node.
# class TreeNode:
#     def __init__(self, val=0, left=None, right=None):
#         self.val = val
#         self.left = left
#         self.right = right
class Solution:
    def allPossibleFBT(self, n: int) -> List[Optional[TreeNode]]:
        @cache
        def dfs(n: int) -> List[Optional[TreeNode]]:
            if n == 1:
                return [TreeNode()]
            ans = []
            for i in range(n - 1):
                j = n - 1 - i
                for left in dfs(i):
                    for right in dfs(j):
                        ans.append(TreeNode(0, left, right))
            return ans

        return dfs(n)

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