Number of Ways to Reorder Array to Get Same BST

This involves understanding how different reorderings of the array will lead to identical binary search trees (BSTs). The solution will involve some intricate recursive logic. We’ll build up to it by outlining the solution approach first:

  1. Understand the Structure of a BST: A BST is built in such a way that the left subtree of a node contains only nodes with keys less than the node’s key, and the right subtree only nodes with keys greater than the node’s key. This provides a unique structure for every unique set of values.

  2. Divide the Problem: We’ll need to break down the problem into subproblems for the left and right subtrees and solve it recursively. For each unique BST, we’ll need to count the number of reorderings of the left and right subtrees that also create identical BSTs.

  3. Use Combinatorics: The solution can be found by multiplying the counts for the left and right subtrees, and then multiplying them by the number of ways the left and right subtrees can be merged. This will involve calculating the binomial coefficient.

  4. Use Memoization: As there will be repeated calculations for the same subarrays, using memoization can greatly speed up the solution.

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from math import factorial

class Solution:
    def numOfWays(self, nums: List[int]) -> int:
        MOD = 10**9 + 7

        def binomial(n, k):
            return factorial(n) // (factorial(k) * factorial(n - k))

        def dfs(nums):
            if len(nums) <= 1:
                return 1
            left = [x for x in nums[1:] if x < nums[0]]
            right = [x for x in nums[1:] if x > nums[0]]
            left_ways = dfs(left)
            right_ways = dfs(right)
            total_ways = left_ways * right_ways * binomial(len(left) + len(right), len(left))
            return total_ways % MOD

        return (dfs(nums) - 1) % MOD
  • binomial(n, k) is a helper function to calculate the binomial coefficient.
  • dfs(nums) is a recursive function that solves the problem for a given subarray of nums. It first determines the values in the left and right subtrees, then finds the number of identical BSTs for the left and right subtrees, and finally calculates the total number of reorderings for the current subarray.
  • The result is reduced by 1 to exclude the original ordering, and then taken modulo 10**9 + 7 as required.

10 Prerequisite LeetCode Problems

“1569. Number of Ways to Reorder Array to Get Same BST” involves a mix of understanding Binary Search Trees (BST), permutations, and dynamic programming. Here are 10 problems to prepare:

  1. “Convert Sorted Array to Binary Search Tree” (LeetCode Problem #108): To start with, it’s good to understand how to construct a BST from an array.

  2. “Validate Binary Search Tree” (LeetCode Problem #98): It’s also crucial to understand properties of BSTs and how to validate them.

  3. “Unique Binary Search Trees” (LeetCode Problem #96): This problem introduces the concept of counting possible BSTs.

  4. “Unique Binary Search Trees II” (LeetCode Problem #95): A follow-up to the above problem, now you need to actually generate all possible BSTs.

  5. “Binary Tree Inorder Traversal” (LeetCode Problem #94): Understanding different traversal methods is important for many tree-based problems.

  6. “Permutations” (LeetCode Problem #46): This problem introduces the concept of generating permutations, which is similar to reordering arrays.

  7. “Permutations II” (LeetCode Problem #47): A follow-up to the previous problem, introducing the concept of dealing with duplicates in permutations.

  8. “Climbing Stairs” (LeetCode Problem #70): This problem is a good starting point for understanding dynamic programming, which might be needed to solve this problem efficiently.

  9. “Unique Paths” (LeetCode Problem #62): A slightly more complex dynamic programming problem involving counting unique paths.

  10. “Unique Paths II” (LeetCode Problem #63): An extension to the Unique Paths problem which introduces obstacles and further refines your dynamic programming skills.

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from math import comb
from typing import List

class Solution:
    def numOfWays(self, nums: List[int]) -> int:
        def f(nums):
            if len(nums) <= 2:
                return 1
            left = [v for v in nums if v < nums[0]]
            right = [v for v in nums if v > nums[0]]
            return comb(len(left) + len(right), len(right)) * f(left) * f(right)

        return (f(nums) - 1) % (10**9 + 7)

Language Agnostic Coding Drills

This Python code defines a class Solution with a method numOfWays. This function uses mathematical combinations and recursive tree structures to calculate the number of binary search trees (BSTs) that can be formed given an array of distinct numbers. It follows these steps:

  1. Importing necessary libraries: The Python standard library has many modules that are helpful for a wide range of tasks. In this case, the math module is imported to use the comb function for calculating combinations.

  2. Basic data types and data structures: Here, the main data structure used is a list of integers, with basic operations such as length calculation, iteration, and list comprehension.

  3. Function definition: Functions are defined to execute a block of code when called. This code defines the function ‘f’.

  4. Conditionals: The ‘if’ statement allows the code to make decisions. In this case, it’s used to check when to stop the recursion.

  5. List Comprehensions: List comprehensions provide a concise way to create lists based on existing lists. Here, it’s used to split the input list into two based on the root of the current BST.

  6. Recursion: Recursion allows a function to call itself. This is used here to calculate the number of BSTs in the left and right subtrees.

  7. Mathematical operations: The comb function calculates the number of ways to choose items from a group without regard to the order of selection.

  8. Class definition: Classes provide a way of bundling data and functionality together. Here, the Solution class is defined.

  9. Modulus operation: The modulus operator (%) returns the remainder of a division. Here, it’s used to prevent the output number from being too large.

Here’s how you can arrange the concepts in increasing level of difficulty:

  1. Basic data types and data structures
  2. Function definition
  3. Conditionals
  4. List comprehensions
  5. Class definition
  6. Importing necessary libraries
  7. Mathematical operations
  8. Recursion
  9. Modulus operation

Targeted Drills in Python

Sure, here are some coding drills based on the concepts mentioned:

  1. Basic data types and data structures:

    • Create a list of integers and perform basic operations like adding an item, removing an item, and calculating the length of the list.
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    nums = [1, 2, 3, 4, 5]
nums.append(6)
nums.remove(1)
length = len(nums)

  2. Function definition:

    • Define a simple function that takes in two numbers and returns their sum.
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    def add(a, b):
    return a + b

  3. Conditionals:

    • Write a function that checks if a number is even or odd.
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    def is_even(n):
    if n % 2 == 0:
        return True
    else:
        return False

  4. List comprehensions:

    • Create a new list that contains only the even numbers from an existing list.
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    nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
even_nums = [num for num in nums if num % 2 == 0]

  5. Class definition:

    • Define a class Person with attributes name and age and a method say_hello.
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    class Person:
    def __init__(self, name, age):
        self.name = name
        self.age = age

    def say_hello(self):
        print(f"Hello, my name is {self.name} and I am {self.age} years old.")

  6. Importing necessary libraries:

    • Import the math module and use it to calculate the square root of a number.
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    import math
sqrt_4 = math.sqrt(4)

  7. Mathematical operations:

    • Use the comb function from the math module to calculate combinations.
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    import math
combinations = math.comb(5, 3)

  8. Recursion:

    • Write a recursive function to calculate the factorial of a number.
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    def factorial(n):
    if n == 1:
        return 1
    else:
        return n * factorial(n - 1)

  9. Modulus operation:

    • Write a function that checks if a number is divisible by another number.
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    def is_divisible(n, m):
    if n % m == 0:
        return True
    else:
        return False

After going through these exercises, one would have a solid understanding of these fundamental concepts, which would then allow them to understand and implement more complex code like the one given in the question.

Problem Classification

Problem Statement: Given an array nums that represents a permutation of integers from 1 to n. We are going to construct a binary search tree (BST) by inserting the elements of nums in order into an initially empty BST. Find the number of different ways to reorder nums so that the constructed BST is identical to that formed from the original array nums.

For example, given nums = [2,1,3], we will have 2 as the root, 1 as a left child, and 3 as a right child. The array [2,3,1] also yields the same BST but [3,2,1] yields a different BST. Return the number of ways to reorder nums such that the BST formed is identical to the original BST formed from nums.

Since the answer may be very large, return it modulo 109 + 7.

Example 1:

Input: nums = [2,1,3] Output: 1 Explanation: We can reorder nums to be [2,3,1] which will yield the same BST. There are no other ways to reorder nums which will yield the same BST. Example 2:

Input: nums = [3,4,5,1,2] Output: 5 Explanation: The following 5 arrays will yield the same BST: [3,1,2,4,5] [3,1,4,2,5] [3,1,4,5,2] [3,4,1,2,5] [3,4,1,5,2] Example 3:

Input: nums = [1,2,3] Output: 0 Explanation: There are no other orderings of nums that will yield the same BST.

Constraints:

1 <= nums.length <= 1000 1 <= nums[i] <= nums.length All integers in nums are distinct.

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Alternatively, if you’re working on a specific problem, you might ask something like:

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Stepwise Refinement

  1. Could you please provide a stepwise refinement of our approach to solving this problem?

  2. How can we take the high-level solution approach and distill it into more granular, actionable steps?

  3. Could you identify any parts of the problem that can be solved independently?

  4. Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Coding Constructs

Consider the following piece of complex software code.

  1. What are the high-level problem-solving strategies or techniques being used by this code?

  2. If you had to explain the purpose of this code to a non-programmer, what would you say?

  3. Can you identify the logical elements or constructs used in this code, independent of any programming language?

  4. Could you describe the algorithmic approach used by this code in plain English?

  5. What are the key steps or operations this code is performing on the input data, and why?

  6. Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

  1. Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.

  2. Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.

  3. Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

  1. Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.

  2. In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.

  3. Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.

Q&A

Similar Problems

Given the problem [provide the problem], identify and list down 10 similar problems on LeetCode. These should cover similar concepts or require similar problem-solving approaches as the provided problem. Please also give a brief reason as to why you think each problem is similar to the given problem.