What is the time complexity of All Possible Full Binary Trees?

The optimal solution has a time complexity of O(4^n / n^(3/2)).

What is the space complexity of All Possible Full Binary Trees?

The space complexity is O(4^n / n^(3/2)).

What data structures are used to solve All Possible Full Binary Trees?

This problem uses Dynamic Programming, Tree, Recursion, Memoization, Binary Tree. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving All Possible Full Binary Trees?

Insight 1: A full binary tree can only have an odd number of nodes. If 'n' is even, there are no possible full binary trees. Insight 2: Recursion is a natural fit for solving this problem. We can break down the problem into smaller subproblems of creating full binary trees with fewer nodes.

Which companies ask LeetCode 894. All Possible Full Binary Trees in interviews?

This problem is frequently asked at Nvidia.

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All Possible Full Binary Trees - LeetCode 894 Solution

Q: How to solve All Possible Full Binary Trees?

The solution uses a recursive approach with memoization. The `solve(k)` function returns a list of all possible full binary trees with 'k' nodes. The base case is when k = 1, where the function returns a list containing a single node with value 0. For larger values of 'k', the function iterates through all possible sizes of the left subtree (i) and calculates the corresponding size of the right subtree (k - 1 - i). It then recursively calls `solve(i)` and `solve(k - 1 - i)` to get the lists of possible left and right subtrees. Finally, it combines each left subtree with each right subtree to create a new full binary tree with 'k' nodes, with the root node having a value of 0.

All Possible Full Binary Trees - Complete Solution Guide

All Possible Full Binary Trees is LeetCode problem 894, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

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. 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,

Detailed Explanation

The problem asks us to generate all possible full binary trees with a given number of nodes 'n'. A full binary tree is a tree where each node has either 0 or 2 children. We are given an integer 'n' representing the number of nodes, and we need to return a list of the root nodes of all possible full binary trees that can be formed using 'n' nodes. Each node in the tree must have a value of 0. The order of the trees in the output list doesn't matter. The constraint is that 1 <= n <= 20.

Solution Approach

The solution uses a recursive approach with memoization. The `solve(k)` function returns a list of all possible full binary trees with 'k' nodes. The base case is when k = 1, where the function returns a list containing a single node with value 0. For larger values of 'k', the function iterates through all possible sizes of the left subtree (i) and calculates the corresponding size of the right subtree (k - 1 - i). It then recursively calls `solve(i)` and `solve(k - 1 - i)` to get the lists of possible left and right subtrees. Finally, it combines each left subtree with each right subtree to create a new full binary tree with 'k' nodes, with the root node having a value of 0.

Step-by-Step Algorithm

Step 1: Check if 'n' is even. If it is, return an empty list because a full binary tree can only have an odd number of nodes.
Step 2: Initialize a memoization table (e.g., a dictionary or map) to store the results of subproblems. The keys are the number of nodes, and the values are the lists of possible full binary trees.
Step 3: Define a recursive function `solve(k)` that takes the number of nodes 'k' as input.
Step 4: In `solve(k)`, first check if the result for 'k' is already stored in the memoization table. If it is, return the stored result.
Step 5: If the result is not in the memoization table, proceed to generate the possible full binary trees for 'k' nodes.
Step 6: Iterate through all possible sizes 'i' of the left subtree, where 'i' ranges from 1 to k-1 with a step of 2 (because the left subtree must also have an odd number of nodes).
Step 7: Calculate the size of the right subtree as k - 1 - i.
Step 8: Recursively call `solve(i)` to get the list of possible left subtrees and `solve(k - 1 - i)` to get the list of possible right subtrees.
Step 9: Iterate through all possible combinations of left and right subtrees, creating a new root node with value 0 and attaching the left and right subtrees to it.
Step 10: Add the new full binary tree to the list of results.
Step 11: Store the list of results in the memoization table for 'k'.
Step 12: Return the list of results.
Step 13: Call `solve(n)` to get the list of all possible full binary trees with 'n' nodes.

Key Insights

Insight 1: A full binary tree can only have an odd number of nodes. If 'n' is even, there are no possible full binary trees.
Insight 2: Recursion is a natural fit for solving this problem. We can break down the problem into smaller subproblems of creating full binary trees with fewer nodes.
Insight 3: Memoization (dynamic programming) is crucial for optimizing the recursive solution. Without memoization, the solution would have exponential time complexity due to overlapping subproblems.

Complexity Analysis

Time Complexity: O(4^n / n^(3/2))

Space Complexity: O(4^n / n^(3/2))

Topics

This problem involves: Dynamic Programming, Tree, Recursion, Memoization, Binary Tree.

Companies

Asked at: Nvidia.

All Possible Full Binary Trees | LeetCode 894

Medium

Dynamic Programming Tree Recursion Memoization Binary Tree

Time: O(4^n / n^(3/2))Space: O(4^n / n^(3/2))

Solution Code

Available in:

/**
 * 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 {
    private Map<Integer, List<TreeNode>> memo = new HashMap<>();
    public List<TreeNode> allPossibleFBT(int n) {
        if (n % 2 == 0) {
            return new ArrayList<>();
        }

        if (!memo.containsKey(1)) {
            List<TreeNode> baseList = new ArrayList<>();
            baseList.add(new TreeNode(0));
            memo.put(1, baseList);
        }


        return solve(n);
    }

    private List<TreeNode> solve(int k) {
        if (memo.containsKey(k)) {
            return memo.get(k);
        }

        List<TreeNode> res = new ArrayList<>();
        for (int i = 1; i < k; i += 2) {
            List<TreeNode> left_subtrees = solve(i);
            List<TreeNode> right_subtrees = solve(k - 1 - i);

            for (TreeNode l : left_subtrees) {
                for (TreeNode r : right_subtrees) {
                    TreeNode root = new TreeNode(0, l, r);
                    res.add(root);
                }
            }
        }

        memo.put(k, res);
        return res;
    }
}

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Problem Info

Description

Problem 894: All Possible Full Binary Trees - Detailed explanation and solution approaches.

Topics

Dynamic Programming Tree Recursion Memoization Binary Tree

Companies

Nvidia

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:894

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

13 steps

1
Step 1: Check if 'n' is even. If it is, return an empty list because a full binary tree can only have an odd number of nodes.
2
Step 2: Initialize a memoization table (e.g., a dictionary or map) to store the results of subproblems. The keys are the number of nodes, and the values are the lists of possible full binary trees.
3
Step 3: Define a recursive function `solve(k)` that takes the number of nodes 'k' as input.
4
Step 4: In `solve(k)`, first check if the result for 'k' is already stored in the memoization table. If it is, return the stored result.
5
Step 5: If the result is not in the memoization table, proceed to generate the possible full binary trees for 'k' nodes.
6
Step 6: Iterate through all possible sizes 'i' of the left subtree, where 'i' ranges from 1 to k-1 with a step of 2 (because the left subtree must also have an odd number of nodes).
7
Step 7: Calculate the size of the right subtree as k - 1 - i.
8
Step 8: Recursively call `solve(i)` to get the list of possible left subtrees and `solve(k - 1 - i)` to get the list of possible right subtrees.
9
Step 9: Iterate through all possible combinations of left and right subtrees, creating a new root node with value 0 and attaching the left and right subtrees to it.
10
Step 10: Add the new full binary tree to the list of results.
11
Step 11: Store the list of results in the memoization table for 'k'.
12
Step 12: Return the list of results.
13
Step 13: Call `solve(n)` to get the list of all possible full binary trees with 'n' nodes.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(4^n / n^(3/2))

💾

Space Complexity

O(4^n / n^(3/2))

📝

Why?

The number of possible Full Binary Trees grows according to the Catalan number. Both the time and space complexities are dependent on the number of possible trees. The memoization avoids recalculating the subproblems, but it needs to store these results, leading to the complexities mentioned.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Bottom-up Dynamic Programming

Instead of using recursion with memoization (top-down DP), we can solve the problem using a bottom-up dynamic programming approach. We start by building the trees for smaller values of 'n' and then use those results to build the trees for larger values of 'n'.

Time: O(4^n / n^(3/2))Space: O(4^n / n^(3/2))

Trade-offs: The bottom-up approach can sometimes be more efficient than the top-down approach due to the elimination of function call overhead. However, the top-down approach is often easier to understand and implement.

✨

Key Insights

1Insight 1: A full binary tree can only have an odd number of nodes. If 'n' is even, there are no possible full binary trees.
2Insight 2: Recursion is a natural fit for solving this problem. We can break down the problem into smaller subproblems of creating full binary trees with fewer nodes.
3Insight 3: Memoization (dynamic programming) is crucial for optimizing the recursive solution. Without memoization, the solution would have exponential time complexity due to overlapping subproblems.

📚

Key Terms Explained

For Students

Dynamic Programming

Breaking complex problems into smaller subproblems and storing results.

Recursion

A function that calls itself to solve smaller versions of the same problem.

Tree

Hierarchical structure with nodes. Like a family tree!

Edge Cases

Edge case 1: n = 1. The solution correctly handles this case by returning a list containing a single node with value 0. This is handled in the base case of the recursive function and the initialization of the memo.
Edge case 2: n is even. The solution returns an empty list, as no full binary tree can be formed with an even number of nodes. This is checked at the beginning of the `allPossibleFBT` function.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Forgetting to memoize the results of subproblems. Without memoization, the solution will have exponential time complexity and will likely time out for larger values of 'n'.

✗Common mistake 2: Not handling the base case (n = 1) correctly. The base case is essential for the recursive algorithm to terminate properly.

✗Common mistake 3: Not checking if 'n' is even before proceeding with the recursive calls. This will lead to incorrect results and potentially infinite recursion.

✗Common mistake 4: Incorrectly calculating the size of the right subtree. The correct formula is k - 1 - i, where k is the total number of nodes and i is the size of the left subtree. The '-1' is to account for the root node.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Compiler design. Parsing expressions can be modeled as tree structures, and generating all possible parse trees with certain constraints is related to this problem.

🏦Real-world scenario 2: Game development. AI decision trees where you may need to explore all possible game states or strategy options.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

Start with recursion, then memoize. The hardest part is identifying the state. Ask yourself: "What information do I need to make the current decision?"

🎯 Interview Tip

Medium problems are the bread and butter of technical interviews. Practice explaining your approach before coding.

⚠️ Watch Out

Forgetting base cases or initializing the DP array incorrectly. Start by solving the smallest subproblems.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: November 22, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJul 13

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 6

❓

Frequently Asked Questions

How do I identify if a problem needs dynamic programming?

DP problems typically have: 1) Overlapping subproblems (same calculation repeated), 2) Optimal substructure (optimal solution uses optimal solutions to subproblems). Look for keywords like "minimum/maximum," "count ways," "is it possible," or "longest/shortest."

How do I choose between recursive and iterative tree traversal?

Use recursive traversal for cleaner code when stack depth isn't a concern (tree height < ~1000). Use iterative with explicit stack for very deep trees to avoid stack overflow. Iterative is also preferred when you need fine-grained control over the traversal process.

What techniques are commonly tested in medium-difficulty problems?

Medium problems often require combining multiple techniques: two-pointer with hash maps, BFS/DFS with state tracking, dynamic programming with optimization, or divide-and-conquer approaches. They test your ability to recognize patterns and choose the right combination of techniques.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "All Possible Full Binary Trees"?

For this medium-level Dynamic Programming problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether tree techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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