##### Problem Statement
Given a binary tree, find the lowest common ancestor (LCA) of two given nodes in the tree. According to the definition of LCA on Wikipedia: “The lowest common ancestor is defined between two nodes `p` and `q` as the lowest node in `T` that has both `p` and `q` as descendants (where we allow a node to be a descendant of itself).”

##### Example
```plaintext
Input: root = [3,5,1,6,2,0,8,null,null,7,4], p = 5, q = 1
Output: 3

Input: root = [3,5,1,6,2,0,8,null,null,7,4], p = 5, q = 4
Output: 5
```

##### Constraints
- The number of nodes in the tree is in the range `[2, 10^5]`.
- `-10^9 <= Node.val <= 10^9`
- All `Node.val` are unique.
- `p != q`
- `p` and `q` will exist in the tree.

#### Expected Challenge Output
When the function `lowestCommonAncestor` is called with the given example inputs, the expected output is the value of the lowest common ancestor of the two nodes.

JavaScript

JavaScript is a high-level, dynamic, and interpreted programming language that is widely used for web development and beyond. It was created in the mid-1990s and has since become one of the most popular and widely used programming languages in the world. JavaScript allows developers to create interactive and dynamic websites, as well as build complex web applications and server-side programs. It is a versatile language that can run on any modern web browser, as well as server environments like Node.js, making it an essential tool for any full-stack web developer.

learn-javascript

Python

Python is a high-level, interpreted, general-purpose programming language. It was created in the late 1980s by Guido van Rossum and has since become one of the most popular programming languages in the world. Python is known for its simple and easy-to-read syntax, making it a great language for beginners to learn, and for experienced developers to write complex applications.

Python is used in a variety of applications, including web development, scientific computing, data analysis, artificial intelligence, and more. It has a large and active community of users and developers who have created many libraries and frameworks that make it easier to build complex applications quickly. Python is also known for its strong support for object-oriented programming and its dynamic typing, which makes it flexible and easy to use.

Overall, Python is a versatile and powerful language that can be used for a wide range of applications, making it a great choice for developers looking to build anything from simple scripts to complex applications.

learn-python

C++ Programming

C++ is a high-level, general-purpose programming language that was created as an extension of the C programming language. It is widely used for developing systems software, game development, and desktop and mobile applications. C++ is known for its performance, efficiency, and versatility, making it a popular choice for developing resource-intensive applications. It is also used in the development of other programming languages, as well as in various scientific and engineering applications. C++ is considered a mid-level language, as it contains both high-level and low-level programming concepts, and provides control over system resources through its use of pointers and manual memory management.

learn-cpp-programming

C Programming

This C fundamentals learning module is designed to teach individuals the basics of programming using the C programming language. The module typically covers the fundamentals of C programming, including topics such as data types, variables, control structures, functions, pointers, and memory management. The module may also cover more advanced topics, such as file handling, dynamic memory allocation, and multithreading.

learn-c-programming

Lowest Common Ancestor in Binary Tree

lowest-common-ancestor-binary-tree

Functions and Recursion

In the cinematic sphere of programming, functions serve as the shooters behind the camera, overseeing and regulating the movie. They play a significant role in breaking down the script (code), making it more manageable and reducing redundancy. Similar to a movie director dictating where the frame begins and ends, a function in C++ dictates how a certain set of instructions will be executed. In this tutorial, we will navigate the labyrinth of C++ functions and dive into the spellbinding magic of recursion.

To begin with, let's understand how functions work. In C++, a function is a block of code that performs a specific task or a group of related tasks. The structure of a function looks like this:

```cpp
<return_type> <function_name>(<parameter_list>){
    // Function Body
}

```

Using a movie analogy, let's create a function named `printActorName`:

```cpp
void printActorName(std::string actorName) {
    std::cout << "Actor's name is: " << actorName << std::endl;
}

```
In this code, `void` is the return type, signaling that our function does not return any value. `printActorName` is the name of our function, and inside the parentheses, we declare a `string` parameter named `actorName`. This function, when called with a string argument, will print that string following the message "Actor's name is:".

We would call the function `printActorName()` in the main function like so:

```cpp
int main() {
    printActorName("Leonardo DiCaprio");
    return 0;
}

```

In the world of programming, where your code can often mirror the complexity of Inception, functions can also return values, much like the totem that lets you differentiate between dreams and reality. Keeping this in mind, let's create a function called `calcMovieDuration` that takes two integers - start time and end time of the movie - and returns the duration.

```cpp
int calcMovieDuration(int startTime, int endTime) {
    int duration = endTime - startTime;
    return duration;
}

```

We would call the `calcMovieDuration()` function like so:

```cpp
int main() {
    int startTime = 9;
    int endTime = 12;
    int duration = calcMovieDuration(startTime, endTime);
    std::cout << "Movie duration is: " << duration << " hours" << std::endl;
    return 0;
}

```

This program finds the difference between the `endTime` and `startTime` and returns it as `duration`. Then, in the `main()` function, the `calcMovieDuration()` function is called with `startTime` and `endTime` as parameters and its returned value is stored in `duration`.

C++ also allows the concept of recursion, akin to our heroes embarking on the inception journey, diving deeper into dreams within dreams. Recursion in programming refers to a function calling itself, creating a loop until a specific condition is met. However, it's essential to define an exit condition, or our heroes would be stuck in a limbo of infinite recursion, much like being trapped in a dream world.

Let's create a function named `countDown` that uses recursion to count down from a given number:

```cpp
void countDown(int num) {
    if (num <= 0) {
       std::cout << "Blast off!" << std::endl;
        return;
    } else {
        std::cout << num << ", ";
        countDown(num-1); // Self calling function or recursive call
    }
}

```

We would call the `countDown()` function like so:

```cpp
int main() {
    countDown(5);
    return 0;
}

```

In the function `countDown`, we're taking an integer as input. If the number is less than or equal to zero, we print "Blast off!" and terminate the function. However, if the number is greater than zero, we print the number, decrement it, and call the `countDown()` function again. This continues until the number is zero, creating a counting down effect.

Functions in C++, just like movie directors, can take complex, convoluted scripts and transform them into beautiful, cohesive stories that are easier to follow and manage. Understanding and correctly using functions can save you from redundancy and unnecessary complexity, thereby making your life, or rather your coding experience, far easier. Recursion might seem like a daunting concept, but with practice and careful application, you can wield this tool like a master, creating highly efficient and effective code.


Dynamic Memory Management

Dynamic Memory Management in C++ is a critical aspect of a developer's toolbox. Typically, when a variable is declared, the compiler allocally determines the amount of memory to be set aside for that variable. When the variable is no longer in use, the set aside memory is reclaimed. However, in certain scenarios, a program might need to allocate memory at runtime. This is where dynamic memory management comes in. This tutorial chapter aims to shed light on dynamic memory management in C++ using various examples.

Unlike [Java](https://stackbay.org/modules/learn-java) where memory allocation and deallocation is handled by the Garbage Collector, in C++, it is the programmer's responsibility to carefully manage dynamic memory, to prevent memory leaks and other related issues.

The keywords 'new' and 'delete' in C++ are used for dynamic memory allocation and deallocation respectively. 

Imagine you are directing a movie named 'MatrixConstruct', and you want to dynamically create new characters during the runtime of the movie as per the script's requirement. Here's how you can do it.

```cpp
// Character structure definition
struct Character {
    std::string name;
    std::string role;

    // Constructor
    Character(std::string name_, std::string role_) : name(name_), role(role_) {}

    // Shows character details
    void showCharacter(){
        cout << "Character Name: " << name << ", Role: " << role << endl;
    }
};

// Current Movie scenario (main)
int main() {
    Character* neo = new Character("Neo", "The One");
    neo->showCharacter();
    // Don't forget to clean up
    delete neo;  
    return 0;
}
```
Here, 'neo' is a pointer to a Character object which is created using 'new'. Once we're done with the 'neo' character, we need to ensure to 'delete' it, else we'll end up with a memory leak.

Now consider a situation where we need to create a collection of characters who form part of an organization named "Zion". Array can be used here, but we only prefer to use arrays when we know the count of characters beforehand. If we need to declare a list of characters at runtime—as in our case—we need to use dynamically allocated arrays.

```cpp
// Create Zion organization characters
int characterCount;
cout << "Enter the number of characters: ";
cin >> characterCount;

Character **zion = new Character*[characterCount];
for(int i = 0; i < characterCount; ++i) {
    string name, role;
    cout << "Enter name for character " << i+1 << ": ";
    cin >> name;
    cout << "Enter role for character " << i+1 << ": ";
    cin >> role;
    zion[i] = new Character(name, role);
}

// Display characters
for(int i = 0; i < characterCount; ++i) {
    zion[i]->showCharacter();
}

// Clean up
for(int i = 0; i < characterCount; ++i) {
    delete zion[i];
}
delete[] zion;

return 0;
```
In the above piece of code, 'zion' is a dynamic array of Character pointers which are each pointing to individual Character objects. Once we're done displaying the characters, we need to loop through the array and 'delete' each character. Lastly, we also need to 'delete' the dynamic array 'zion' itself.

Always remember, for every 'new', there should be a 'delete'. It is pivotal to deallocate or 'delete' any memory that was previously allocated with 'new'. Failing to do so will cause a memory leak, wherein the memory space remains occupied but cannot be utilized or recovered until the program terminates. In large scale software or long-running systems, this could lead to significant issues as the system can run out of memory.

Understanding dynamic memory management in C++ is crucial, especially for resource-constrained environments and when dealing with large amounts of data. It provides developers with the power to manage system resources efficiently, at the cost of adding manual memory management tasks. Moreover, it also enables more complex data structures like linked lists and binary trees, which can be dynamically enlarged or shrunk, allowing for highly adaptable and efficient software solutions. 

In a future chapter, we will explore more complex uses of dynamic memory through data structures and more real-world examples. For now, practice remembering the 'new' must always be paired with 'delete', and understanding how dynamic allocation is often crucial to manage complex, flexible data structures, and ultimately, creating an epic adventure like 'MatrixConstruct'.

Recursion

Recursion is a programming paradigm that involves solving a problem by breaking it down into smaller sub-problems of the same kind. This approach can be highly effective when dealing with complex problems that require repeated calculations or processing. In [JavaScript](https://stackbay.org/modules/learn-javascript), recursion is implemented using functions that call themselves repeatedly until a base case is reached.

Recursion is not a new concept in programming, and it has been around for decades. However, the rise of [JavaScript](https://stackbay.org/modules/learn-javascript) as a popular programming language has made it an increasingly popular method for solving complex problems. Recursive programming in [JavaScript](https://stackbay.org/modules/learn-javascript) can be highly effective when dealing with large datasets or complex algorithms that require repeated calculations.

In this article, we will explore what recursion is and how to use it effectively in [JavaScript](https://stackbay.org/modules/learn-javascript). We will also cover some common use cases for recursion in [JavaScript](https://stackbay.org/modules/learn-javascript) and provide examples of code to illustrate our points.

##### What is Recursion?

Recursion is the process of a function calling itself repeatedly until a base case is reached. A base case is a condition that causes the function to terminate without further calls to itself. The key idea behind recursion is that a complex problem can be broken down into smaller sub-problems that are easier to solve.

Recursion can be used in many different programming languages, including [JavaScript](https://stackbay.org/modules/learn-javascript). In [JavaScript](https://stackbay.org/modules/learn-javascript), a recursive function looks very similar to any other function, with the exception of one important difference: it calls itself. A common example of a recursive function is the factorial function.

##### The Factorial Function

The factorial function is a mathematical function that takes an integer as input and returns its factorial value. The factorial of a number n is defined as the product of all positive integers from 1 to n. For example, the factorial of 5 (written as 5!) is equal to 5 x 4 x 3 x 2 x 1 = 120.

To calculate the factorial of a number in [JavaScript](https://stackbay.org/modules/learn-javascript) using recursion, we can define a function that takes an integer n and calls itself repeatedly until it reaches a base case:

```js
function factorial(n) {
  if (n === 0) {
    return 1; // base case: the factorial of 0 is 1
  } else {
    return n * factorial(n - 1); // recursive call
  }
}
```

In this code, we define a function called `factorial` that takes an integer as input. If the number is equal to 0 (the base case), we return 1. Otherwise, we use recursion to calculate the factorial of n - 1 and multiply it by n. This process continues until we reach the base case, at which point the function stops calling itself and returns the final result.

##### Another Example: Finding the Length of a List

Recursion can be used in many different ways to solve problems in [JavaScript](https://stackbay.org/modules/learn-javascript). For example, let's say we want to find the length of a list without using a for loop or any other iterative method. We could use recursion to calculate the length of the list by calling itself repeatedly until it reaches an empty array (the base case).

Here is an example of how to use recursion to find the length of a list in [JavaScript](https://stackbay.org/modules/learn-javascript):

```js
function length(arr) {
  if (arr.length === 0) { // base case: an empty array has a length of 0
    return 0;
  } else {
    return 1 + length(arr.slice(1)); // recursive call with sliced array
  }
}
```

In this code, we define a function called `length` that takes an array as input. If the array is empty (the base case), we return 0. Otherwise, we use recursion to calculate the length of the array by slicing off the first element and calling `length` on the remaining elements in the array. This process continues until we reach the base case, at which point the function stops calling itself and returns the final result.

##### Common Use Cases for Recursion in [JavaScript](https://stackbay.org/modules/learn-javascript)

Recursion can be used to solve a wide variety of problems in [JavaScript](https://stackbay.org/modules/learn-javascript). Some common use cases for recursion include:

- Sorting algorithms: Many sorting algorithms, such as quicksort or mergesort, use recursion to divide the input data into smaller subsets that are easier to sort.
- Tree traversal: Recursion can be used to traverse trees and perform operations on each node in the tree.
- Depth-first search: Recursion is often used to implement depth-first search algorithms, which explore all possible paths in a graph or tree before backtracking.
- Generating Fibonacci numbers: The Fibonacci sequence can be generated using recursion by calling itself repeatedly with different arguments.

In conclusion, recursion is an important programming paradigm that can help us solve complex problems using functions that call themselves repeatedly until a base case is reached. While recursive programming can be more difficult to understand than iterative programming, it offers many advantages for solving certain types of problems in [JavaScript](https://stackbay.org/modules/learn-javascript). By mastering the art of recursion, we can become better programmers and tackle more challenging problems with ease.

Lowest Common Ancestor in Binary Tree

Problem Statement

Example

Constraints

Expected Challenge Output

Test Cases