Data Structures

January 17, 2023

This article was originally written by James Kyle and posted on GitHub. It has been posted here to make it easier to follow.

Today we're gonna learn all about data structures. "OOooooOOOooh *soexciting" right? Yeah, they definitely aren't the juiciest topic out there, but they are important. Not just to pass computer science 101, but in order to be a better programmer.

• Manage complexity and make your programs easier to follow.
• Build high-performance, memory-efficient programs.

I believe that the former is more important. Using the right data structure can drastically simplify what would otherwise be complicated logic. The latter is important too. If performance or memory matters then using the right data structure is more than often essential.

In order to learn about data structures, we're going to implement a few of them together. Don't worry, we'll keep the code nice and short.

## What are Data Structures?

Data Structures are the in-memory representation of the data. They are the way in which we store and organize data in a computer.

Essentially, they are different methods of storing and organizing data that serve a number of different needs. Data can always be represented in many different ways. However, depending on what that data is and what you need to do with it, one representation will be a better choice than the others. To understand why let's first talk a bit about algorithms.

## Algorithms

An algorithm is a set of instructions to solve a problem. It is a step-by-step procedure that takes some input and produces some output.

Data structures are implemented with algorithms, and algorithms are implemented with data structures. It's data structures and algorithms all the way down until you reach the microscopic people with punch cards that control the computer. (That's how computers work right?)

Any given task can be implemented in an infinite number of ways. So for common tasks there are often many different algorithms that people have come up with. For example, there are an absurd number of algorithms for sorting a set of unordered items:

• Insertion Sort
• Selection Sort
• Merge Sort
• Bubble Sort
• Heap Sort
• Quick Sort
• Shell Sort
• Timsort
• Bucket Sort
• ...etc.

Some of these are significantly faster than others. Some use less memory. Some are easy to implement. Some are based on assumptions about the dataset. Every single one of them will be better for something. So you'll need to make a decision based on what your needs are and for that, you'll need a way of comparing them, a way to measure them.

When we compare the performance of algorithms we use a rough measurement of their average and worst-case performance using something called "Big-O".

## Big-O Notation

Big-O Notation is a way of roughly measuring the performance of algorithms in order to compare one against another when discussing them.

Big-O is a mathematical notation that we borrowed in computer science to classify algorithms by how they respond to the number (N) of items that you give them. There are two primary things that you measure with Big-O:

• Time complexity refers to the total count of operations an algorithm will perform given a set of items.
• Space complexity refers to the total memory an algorithm will take up while running given a set of items.

We measure these independently from one another because while an algorithm may perform fewer operations than another, it may also take up way more memory. Depending on what your requirements are, one may be a better choice than the other.

These are some common Big-O's from best to worst:

`````` Name           Notation     How you feel when they show up at your party
------------------------------------------------------------------------
Constant       O(1)         AWESOME!!
Logarithmic    O(log N)     GREAT!
Linear         O(N)         OKAY.
Linearithmic   O(N log N)   UGH...
Polynomial     O(N ^ 2)     SHITTY
Exponential    O(2 ^ N)     HORRIBLE
Factorial      O(N!)        WTF``````

To give you an idea of how many operations we're talking about. Let's look at what these would equal given the (N) number of items.

``````            N = 5            10             20             30
-----------------------------------------------------------------------
O(1)           1            1              1              1
O(log N)       2.3219...    3.3219...      4.3219...      4.9068...
O(N)           5            10             20             30
O(N log N)     11.609...    33.219...      84.638...      147.204...
O(N ^ 2)       25           100            400            900
O(2 ^ N)       32           1024           1,048,576      1,073,741,824
O(N!)          120          3,628,800      2,432,902,0... 265,252,859,812,191,058,636,308,480,000,000``````

As you can see, even for relatively small sets of data you could do a lot of extra work.

With data structures, you can perform 4 primary types of actions: Accessing, Searching, Inserting, and Deleting.

It is important to note that data structures may be good at one action but bad at another.

``````                        Accessing    Searching    Inserting    Deleting
-------------------------------------------------------------------------
Array     O(1)         O(N)         O(N)         O(N)
Linked List     O(N)         O(N)         O(1)         O(1)
Binary Search Tree     O(log N)     O(log N)     O(log N)     O(log N)``````

Or rather...

``````                        Accessing    Searching    Inserting    Deleting
-------------------------------------------------------------------------
Array     AWESOME!!    OKAY         OKAY         OKAY
Linked List     OKAY         OKAY         AWESOME!!    AWESOME!!
Binary Search Tree     GREAT!       GREAT!       GREAT!       GREAT!``````

Even further, some actions will have a different "average" performance and a "worst case scenario" performance.

There is no perfect data structure, and you choose one over another based on the data that you are working with and the things you are going to do with it. This is why it is important to know a number of different common data structures so that you can choose from them.

## Memory

A computer's memory is pretty boring, it's just a bunch of ordered slots where you can store information. You hold onto memory addresses in order to find information.

Let's imagine a chunk of memory like this:

``````   Values: |1001|0110|1000|0100|0101|1010|0010|0001|1101|1011...
Addresses: 0    1    2    3    4    5    6    7    8    9    ...``````

If you've ever wondered why things are zero-indexed in programming languages before, it is because of the way that memory works. If you want to read the first chunk of memory you read from 0 to 1, the second you read from 1 to 2. So the address that you hold onto for each of those is 0 and 1 respectively.

Your computer has much much more memory than this, and it is all just a continuation of the pattern above.

Memory is a bit like the wild west, every program running on your machine is stored within this same physical data structure. Without layers of abstraction over it, it would be extremely difficult to use.

But these abstractions serve two additional purposes:

• Storing data in memory in a way that is more efficient and/or faster to work with.
• Storing data in memory in a way that makes it easier to use.

## Lists

To demonstrate the raw interaction between memory and a data structure we're going to first implement a list.

A list is a representation of an ordered sequence of values where the same value may appear many times.

``````class List {
/**
* We start with an empty block of memory which we are going to represent
* with a normal JavaScript array and we'll store the length of the list.
*
* Note that we want to store the length separately because in real life the
* "memory" doesn't have a length you can read from.
*/
constructor() {
this.memory = [];
this.length = 0;
}

/**
* First we need a way to retrieve data from our list.
*
* With a plain list, you have very fast memory access because you keep track
*
* List access is constant O(1) - "AWESOME!!"
*/
}

/**
* Because lists have an order you can insert stuff at the start, middle,
* or end of them.
*
* For our implementation, we're going to focus on adding and removing values
* at the start or end of our list with these four methods:
*
*   - Push    - Add value to the end
*   - Pop     - Remove a value from the end
*   - Unshift - Add value to the start
*   - Shift   - Remove a value from the start
*/

/*
* Starting with "push" we need a way to add items to the end of the list.
*
* It is as simple as adding a value in the address after the end of our
* list. Because we store the length this is easy to calculate. We just add
* the value and increment our length.
*
* Pushing an item to the end of a list is constant O(1) - "AWESOME!!"
*/
push(value) {
this.memory[this.length] = value;
this.length++;
}

/**
* Next we need a way to "pop" items off of the end of our list.
*
* Similar to push all we need to do is remove the value at the address at
* the end of our list. Then just decrement length.
*
* Popping an item from the end of a list is constant O(1) - "AWESOME!!"
*/
pop() {
// Don't do anything if we don't have any items.
if (this.length === 0) return;

// Get the last value, stop storing it, and return it.
let lastAddress = this.length - 1;
this.length--;

// Also return the value so it can be used.
return value;
}

/**
* "push" and "pop" both operate on the end of a list, and overall are pretty
* simple operations because they don't need to be concerned with the rest of
* the list.
*
* Let's see what happens when we operate at the beginning of the list with
* "unshift" and "shift".
*/

/**
* In order to add a new item at the beginning of our list, we need to make
* room for our value at the start by sliding all of the values over by one.
*
*     [a, b, c, d, e]
*      0  1  2  3  4
*       ⬊  ⬊  ⬊  ⬊  ⬊
*         1  2  3  4  5
*     [x, a, b, c, d, e]
*
* In order to slide all of the items over we need to iterate over each one
* moving the prev value over.
*
* Because we have to iterate over every single item in the list:
*
* Unshifting an item to the start of a list is linear O(N) - "OKAY."
*/
unshift(value) {
// Store the value we are going to add to the start.
let previous = value;

// Iterate through each item...
// replacing the "current" value with the "previous" value and storing the
// "current" value for the next iteration.
previous = current;
}

// Add the last item in a new position at the end of the list.
this.memory[this.length] = previous;
this.length++;
}

/**
* Finally, we need to write a shift function to move in the opposite
* direction.
*
* We delete the first value and then slide through every single item in the
* list to move it down one address.
*
*     [x, a, b, c, d, e]
*         1  2  3  4  5
*       ⬋  ⬋  ⬋  ⬋  ⬋
*      0  1  2  3  4
*     [a, b, c, d, e]
*
* Shifting an item from the start of a list is linear O(N) - "OKAY."
*/
shift() {
// Don't do anything if we don't have any items.
if (this.length === 0) return;

let value = this.memory;

// Iterate through each item...
// and replace them with the next item in the list.
}

// Delete the last item since it is now in the previous address.
delete this.memory[this.length - 1];
this.length--;

return value;
}
}``````

Lists are great for fast access and dealing with items at the end. However, as we've seen it isn't great at dealing with items not at the end of the list and we have to manually hold onto memory addresses.

So let's take a look at a different data structure and how it deals with adding, accessing, and removing values without needing to know memory addresses.

## Hash Tables

A hash table is a data structure that's unordered. Instead we have "keys" and "values" where we computed an address in memory using the key.

The basic idea is that we have keys that are "hashable" (which we'll get to in a second) and can be used to add, access, and remove from memory very efficiently.

``````const hashTable = new HashTable();

hashTable.set('myKey', 'myValue');
hashTable.get('myKey'); // >> 'myValue'``````

Let's take a look at how we can implement a hash table.

``````class HashTable {

/**
* Again we're going to use a plain JavaScript array to represent our memory.
*/
constructor() {
this.memory = [];
}

/**
* In order to store key-value pairs in memory from our hash table we need a
* way to take the key and turn it into an address. We do this through an
* operation known as "hashing".
*
* Basically it takes a key and serializes it into a unique number for that
* key.
*
*    hashKey("abc") =>  96354
*    hashKey("xyz") => 119193
*
* You have to be careful though, if you had a really big key you don't want
* to match it to a memory address that does not exist.
*
* So the hashing algorithm needs to limit the size, which means that there
* are a limited number of addresses for an unlimited number of values.
*
* The result is that you can end up with collisions. Places where two keys
* get turned into the same address.
*
* Any real-world hash table implementation would have to deal with this,
* however, we are just going to glaze over it and pretend that doesn't happen.
*/

/**
* Let's set up our "hashKey" function.
*
* Don't worry about understanding the logic of this function, just know that
* it accepts a string and outputs a (mostly) unique address that we will use
* in all of our other functions.
*/
hashKey(key) {
let hash = 0;
for (let index = 0; index < key.length; index++) {
// Oh look– magic.
let code = key.charCodeAt(index);
hash = ((hash << 5) - hash) + code | 0;
}
return hash;
}

/**
* Next, let's define our "get" function so we have a way of accessing values
* by their key.
*
* HashTable access is constant O(1) - "AWESOME!!"
*/
get(key) {
// We start by turning our key into an address.
// Then we simply return whatever is at that address.
}

/**
* We also need a way of adding data before we access it, so we will create
* a "set" function that inserts values.
*
* HashTable setting is constant O(1) - "AWESOME!!"
*/
set(key, value) {
// Again we start by turning the key into an address.
// Then just set the value at that address.
}

/**
* Finally we just need a way to remove items from our hash table.
*
* HashTable deletion is constant O(1) - "AWESOME!!"
*/
remove(key) {
// As always, we hash the key to get an address.
// Then, if it exists, we `delete` it.
}
}
}``````

From this point going forward we are going to stop interacting directly with memory as the rest of these data structures start to be implemented with other data structures.

These data structures focus on doing two things:

• Organizing data based on how it is used
• Abstracting away implementation details

These data structures focus on creating an organization that makes sense for various types of programs. They insert a language that allows you to discuss more complicated logic. All of this while abstracting away implementation details so that their implementation can change to be made faster.

## Stacks

Stacks are similar to lists in that they have an order, but they limit you to only pushing and popping values at the end of the list, which as we saw before are very fast operations when mapping directly to memory.

However, Stacks can also be implemented with other data structures in order to add functionality to them.

The most common usage of the stacks is in the places where you have one process adding items to the stack and another process removing them from the end– prioritizing items added most recently.

``````class Stack {

/**
* We're going to again be backed by a JavaScript array, but this time it
* represents a list like we implemented before rather than memory.
*/
constructor() {
this.list = [];
this.length = 0;
}

/**
* We're going to implement two of the functions from list's "push" and "pop"
* which are going to be identical in terms of functionality.
*/

/**
* Push to add items to the top of the stack.
*/
push(value) {
this.length++;
this.list.push(value);
}

/**
* And pop to remove items from the top of the stack.
*/
pop() {
// Don't do anything if we don't have any items.
if (this.length === 0) return;

// Pop the last item off the end of the list and return the value.
this.length--;
return this.list.pop();
}

/**
* We're also going to add a function in order to view the item at the top of
* the stack without removing it from the stack.
*/
peek() {
// Return the last item in "items" without removing it.
return this.list[this.length - 1];
}
}``````

## Queues

Next, we're going to build a queue which is complementary to stacks. The difference is that this time you remove items from the start of the queue rather than the end. Removing the oldest items rather than the most recent.

Again, because this limits the amount of functionality, there are many different ways of implementing it. A good way might be to use a linked list which we will see later.

``````class Queue {
/**
* Again, our queue is using a JavaScript array as a list rather than memory.
*/
constructor() {
this.list = [];
this.length = 0;
}

/**
* Similar to stacks we're going to define two functions for adding and
* removing items from the queue. The first is "enqueue".
*
* This will push values to the end of the list.
*/
enqueue(value) {
this.length++;
this.list.push(value);
}

/**
* Next is "dequeue", instead of removing the item from the end of the list,
* we're going to remove it from the start.
*/
dequeue() {
// Don't do anything if we don't have any items.
if (this.length === 0) return;

// Shift the first item off the start of the list and return the value.
this.length--;
return this.list.shift();
}

/**
* Same as stacks we're going to define a "peek" function for getting the next
* value without removing it from the queue.
*/
peek() {
return this.list;
}
}``````

The important thing to note here is that because we used a list to back our queue it inherits the performance of "shift" which is linear O(N) "OKAY."

Later we'll see linked lists that will allow us to implement a much faster Queue.

From this point forward we're going to start dealing with data structures where the values of the data structure reference one another.

``````+- Data Structure ---------------------------------------+
|  +- Item A ---------------+ +- Item B ---------------+ |
|  | Value: 1               | | Value: 2               | |
|  | Reference to: (Item B) | | Reference to: (Item A) | |
|  +------------------------+ +------------------------+ |
+--------------------------------------------------------+``````

The values inside the data structure become their own mini data structures in that they contain a value along with additional information including references to other items within the overall data structure.

You'll see what I mean by this in a second.

## Graphs

A graph is not a visual chart of some sort. It is a data structure that consists of nodes and edges. Nodes are the values that are stored in the graph and edges are the connections between the nodes. Imagine a map of a city. Each intersection is a node and the roads are the edges. You can travel from one node to another by following the edges.

Imagine the following graph:

``````A –→ B ←–––– C → D ↔ E
↑    ↕     ↙ ↑     ↘
F –→ G → H ← I ––––→ J
↓     ↘ ↑
K       L``````

We have a bunch of "nodes" (A, B, C, D, ...) that are connected with lines. These nodes are going to look like this:

``````Node {
value: ...,
lines: [(Node), (Node), ...]
}``````

The entire graph will look like this:

``````Graph {
nodes: [
Node {...},
Node {...},
...
]
}``````

Alright, let's implement this.

``````class Graph {
/**
* We'll hold onto all of our nodes in a regular JavaScript array. Not
* because there is any particular order to the nodes but because we need a
* way to store references to everything.
*/
constructor() {
this.nodes = [];
}

/**
* We can start to add values to our graph by creating nodes without any
* lines.
*/
return this.nodes.push({
value,
lines: []
});
}

/**
* Next we need to be able to lookup nodes in the graph. Most of the time
* you'd have another data structure on top of a graph in order to make
* searching faster.
*
* But for our case, we're simply going to search through all of the nodes to find
* the one with the matching value. This is a slower option, but it works for
* now.
*/
find(value) {
return this.nodes.find(node => {
return node.value === value;
});
}

/**
* Next we can connect two nodes by making a "line" from one to the other.
*/
// Find the nodes for each value.
let startNode = this.find(startValue);
let endNode = this.find(endValue);

// Freak out if we didn't find one or the other.
if (!startNode || !endNode) {
throw new Error('Both nodes need to exist');
}

// And add a reference to the endNode from the startNode.
startNode.lines.push(endNode);
}
}``````

Finally you can use a graph like this:

``````const graph = new Graph();

const two = graph.find(1).lines;``````

This might seem like a lot of work to do very little, but it's actually a quite powerful pattern, especially for finding sanity in complex programs.

They do this by optimizing for the connections between data rather than operating on the data itself. Once you have one node in the graph, it's extremely simple to find all the related items in the graph.

Tons of things can be represented this way, users with friends, the 800 transitive dependencies in a node_modules folder, the internet itself is a graph of webpages connected together by links.

Next we're going to see how a graph-like structure can help optimize ordered lists of data.

Linked lists are a very common data structure that is often used to implement other data structures because of its ability to efficiently add items to the start, middle, and end.

The basic idea of a linked list is similar to a graph. You have nodes that point to other nodes. They look sorta like this:

``1 -> 2 -> 3 -> 4 -> 5``

Visualizing them as a JSON-like structure looks like this:

``````{
value: 1,
next: {
value: 2,
next: {
value: 3,
next: {...}
}
}
}``````

Alright, let's implement this.

``````class LinkedList {

/**
* Unlike a graph, a linked list has a single node that starts off the entire
* chain. This is known as the "head" of the linked list.
*
* We're also going to track the length.
*/
constructor() {
this.length = 0;
}

/**
* First we need a way to retrieve a value in a given position.
*
* This works differently than normal lists as we can't just jump to the
* correct position. Instead, we need to move through the individual nodes.
*/
get(position) {
// Throw an error if position is greater than the length of the LinkedList
if (position >= this.length) {
throw new Error('Position outside of list range');
}

// Slide through all of the items using node.next until we reach the
// specified position.
for (let index = 0; index < position; index++) {
current = current.next;
}

// Return the node we found.
return current;
}

/**
* Next we need a way to add nodes to the specified position.
*
* We're going for a generic add method that accepts a value and a position.
*/
// First create a node to hold our value.
let node = {
value,
next: null
};

// We need to have a special case for nodes being inserted at the head.
// We'll set the "next" field to the current head and then replace it with
// our new node.
if (position === 0) {

// If we're adding a node in any other position we need to splice it in
// between the current node and the previous node.
} else {
// First, find the previous node and the current node.
let prev = this.get(position - 1);
let current = prev.next;
// Then insert the new node in between them by setting its "next" field
// to the current node and updating the previous node's "next" field to
// the new one.
node.next = current;
prev.next = node;
}

// Finally just increment the length.
this.length++;
}

/**
* The last method we need is a remove method. We're just going to look up a
* node by its position and splice it out of the chain.
*/
remove(position) {
// We should not be able to remove from an empty list
throw new Error('Removing from empty list')
}
// If we're removing the first node we simply need to set the head to the
// next node in the chain
if (position === 0) {

// For any other position, we need to look up the previous node and set it
// to the node after the current position.
} else {
let prev = this.get(position - 1);
prev.next = prev.next.next;
}

// Then we just decrement the length.
this.length--;
}
}``````

The remaining two data structures we are going to cover are both in the "tree" family. Much like real life, there are many different types of tree data structures.

• Binary Trees: AA Tree, AVL Tree, Binary Search Tree, Binary Tree, Cartesian Tree, left child/right sibling tree, order statistic tree, Pagoda, etc
• B Trees: B Tree, B+ Tree, B* Tree, B Sharp Tree, Dancing Tree, 2-3 Tree, etc
• Heaps: Heap, Binary Heap, Weak Heap, Binomial Heap, Fibonacci Heap, Leonardo, Heap, 2-3 Heap, Soft Heap, Pairing Heap, Leftist Heap, Treap etc
• Trees: Trie, Radix Tree, Suffix Tree, Suffix Array, FM-index, B-trie etc
• Multi-way Trees: Ternary Tree, K-ary tree, And-or tree, (a,b)-tree, Link/Cut Tree etc
• Space Partitioning Trees: Segment Tree, Interval Tree, Range Tree, Bin, Kd Tree, Quadtree, Octree, Z-Order, UB-Tree, R-Tree, X-Tree, Metric Tree, Cover Tree, etc
• Application-Specific Tree: Abstract Syntax Tree, Parse Tree, Decision Tree, Minimax Tree etc

Little did you know you'd be studying dendrology today... and that's not even all of them. But don't let any of this scare you, most of those don't matter at all. There were just a lot of Computer Science PhDs who had something to prove.

Trees are much like graphs or linked lists except they are "unidirectional". All this means is that they can't have loops of references.

``````  Tree:                Not a Tree:

A                        A
↙   ↘                    ↗   ↘
B       C                B ←–––– C``````

If you can draw a loop between connected nodes in a tree... well, you don't have a tree.

Trees have many different uses, they can be used to optimize searching or sorting. They can organize programs better. They can give you a representation that is easier to work with.

## Trees

We'll start off with an extremely simple tree structure. It doesn't have any special rules to it and looks something like this:

``````Tree {
root: {
value: 1,
children: [{
value: 2,
children: [...]
}, {
value: 3,
children: [...]
}]
}
}``````

Alright, let's see how we can implement this.

``````class Tree {

/**
* The tree has to start with a single parent, the "root" of the tree.
*/
constructor() {
this.root = null;
}

/**
* We need a way to traverse our tree and call a function on each node in the
* tree.
*/
traverse(callback) {
// We'll define a walk function that we can call recursively on every node
// in the tree.
function walk(node) {
// First call the callback on the node.
callback(node);
// Then recursively call the walk function on all of its children.
node.children.forEach(walk);
}

// Now kick the traversal process off.
walk(this.root);
}

/**
* Next we need a way to add nodes to our tree.
*/
let newNode = {
value,
children: []
};

// If there is no root, just set it to the new node.
if (this.root === null) {
this.root = newNode;
return;
}

// Otherwise traverse the entire tree and find a node with a matching value
// and add the new node to its children.
this.traverse(node => {
if (node.value === parentValue) {
node.children.push(newNode);
}
});
}
}``````

This is one of the most basic trees you could have and is probably only useful if the data you are representing actually resembles a tree. But with some extra rules, a tree can serve a lot of different purposes.

## Binary Search Trees

Binary search trees are a fairly common form of tree for their ability to efficiently access, search, insert, and delete values all while keeping them in a sorted order.

Imagine taking a sequence of numbers:

``1  2  3  4  5  6  7``

And turning it into a tree starting from the center.

``````           4
/     \
2           6
/   \       /   \
1     3     5     7
-^--^--^--^--^--^--^-
1  2  3  4  5  6  7``````

This is how a binary tree works. Each node can have two children:

• Left: Less than parent node's value.
• Right: Greater than parent node's value.

Note: In order to make this work all values must be unique in the tree.

This makes the traversal to find a value very efficient. Say we're trying to find the number 5 in our tree:

``````        (4)         <--- 5 > 4, so move right.
/     \
2         (6)    <--- 5 < 6, so move left.
/   \       /   \
1     3    (5)    7 <--- We've reached 5!``````

Notice how we only had to do 3 checks to reach the number 5. If we were to expand this tree to 1000 items. We'd go:

`` 500 -> 250 -> 125 -> 62 -> 31 -> 15 -> 7 -> 3 -> 4 -> 5``

Only 10 checks for 1000 items!

The other important thing about binary search trees is that they are similar to linked lists in the sense that you only need to update the immediately surrounding items when adding or removing a value.

``````class BinarySearchTree {

/**
* Same as the previous Tree, we need to have a "root" of the binary search
* tree.
*/
constructor() {
this.root = null;
}

/**
* In order to test if the value exists in the tree, we first need to search
* through the tree.
*/
contains(value) {
// We start at the root.
let current = this.root;

// We're going to keep running as long as we have another node to visit.
// If we reach a `left` or `right` that is `null` then this loop ends.
while (current) {

// If the value is greater than the current.value we move to the right
if (value > current.value) {
current = current.right;

// If the value is less than the current.value we move to the left.
} else if (value < current.value) {
current = current.left;

// Otherwise we must be equal values and we return true.
} else {
return true;
}
}

// If we haven't matched anything then we return false.
return false;
}

/**
* In order to add items to this tree we are going to do the same traversal
* as before, bouncing between left and right nodes depending on them being
* less than or greater than the value we're adding.
*
* However, this time when we reach a `left` or `right` that is `null` we're
* going to add a new node in that position.
*/
// First let's setup our node.
let node = {
value: value,
left: null,
right: null
};

// Special case for when there isn't any root node and we just need to add
// one.
if (this.root === null) {
this.root = node;
return;
}

// We start at the root.
let current = this.root;

// We're going to loop until we've either added our item or discovered it
// already exists in the tree.
while (true) {

// If the value is greater than the current.value we move to the right.
if (value > current.value) {

// If `right` does not exist, set it to our node, and stop traversing.
if (!current.right) {
current.right = node;
break;
}

// Otherwise just move on to the right node.
current = current.right;

// If the value is less than the current.value we move to the left.
} else if (value < current.value) {

// If `left` does not exist, set it to our node, and stop traversing.
if (!current.left) {
current.left = node;
break;
}

// Otherwise just move on to the left node.
current = current.left;

// If the number isn't less than or greater, then it must be the same and
// we don't do anything.
} else {
break;
}
}
}
}``````

## The End

I know that was probably a bit dense, but hopefully it gave you a good amount of knowledge. If you enjoyed it, would you mind giving the repo a star and follow me on twitter @buildsghost.