Everything you need to know about tree data structures
TK Dec 6 '18 ・1 min read
When you first learn to code, it’s common to learn arrays as the “main data structure.”
Eventually, you will learn about hash tables
too. If you are pursuing a Computer Science degree, you have to take a class on data structure. You will also learn about linked lists
, queues
, and stacks
. Those data structures are called “linear” data structures because they all have a logical start and a logical end.
When we start learning about trees
and graphs
, it can get really confusing. We don’t store data in a linear way. Both data structures store data in a specific way.
This post is to help you better understand the Tree Data Structure and to clarify any confusion you may have about it.
In this article, we will learn:
What is a tree
Examples of trees
Its terminology and how it works
How to implement tree structures in code.
Let’s start this learning journey. :)
Definition
When starting out programming, it is common to understand better the linear data structures than data structures like trees and graphs.
Trees are wellknown as a nonlinear data structure. They don’t store data in a linear way. They organize data hierarchically.
Let’s dive into real life examples!
What do I mean when I say in a hierarchical way?
Imagine a family tree with relationships from all generation: grandparents, parents, children, siblings, etc. We commonly organize family trees hierarchically.
The above drawing is is my family tree. Tossico, Akikazu, Hitomi,
and Takemi
are my grandparents.
Toshiaki
and Juliana
are my parents.
TK, Yuji, Bruno
, and Kaio
are the children of my parents (me and my brothers).
An organization’s structure is another example of a hierarchy.
In HTML, the Document Object Model (DOM) works as a tree.
The HTML
tag contains other tags. We have a head
tag and a body
tag. Those tags contains specific elements. The head
tag has meta
and title
tags. The body
tag has elements that show in the user interface, for example, h1
, a
, li
, etc.
A technical definition
A tree
is a collection of entities called nodes
. Nodes are connected by edges
. Each node
contains a value
or data
, and it may or may not have a child node
.
The first node
of the tree
is called the root
. If this root node
is connected by another node
, the root
is then a parent node
and the connected node
is a child
.
AllTree nodes
are connected by links called edges
. It’s an important part of trees
, because it’s manages the relationship between nodes
.
Leaves
are the last nodes
on a tree.
They are nodes without children. Like real trees, we have the root
, branches
, and finally the leaves
.
Other important concepts to understand are height
and depth
.
The height
of a tree
is the length of the longest path to a leaf
.
The depth
of a node
is the length of the path to its root
.
Terminology summary
**Root **is the topmost
node
of thetree
**Edge **is the link between two
nodes
**Child **is a
node
that has aparent node
**Parent **is a
node
that has anedge
to achild node
**Leaf **is a
node
that does not have achild node
in thetree
**Height **is the length of the longest path to a
leaf
**Depth **is the length of the path to its
root
Binary trees
Now we will discuss a specific type of tree
. We call it thebinary tree
.
“In computer science, a binary tree is a tree data structure in which each node has at the most two children, which are referred to as the left child and the right child.” — Wikipedia
So let’s look at an example of a binary tree
.
Let’s code a binary tree
The first thing we need to keep in mind when we implement a binary tree
is that it is a collection of nodes
. Each node
has three attributes: value
, left_child
, and right_child
.
How do we implement a simple binary tree
that initializes with these three properties?
Let’s take a look.
class BinaryTree:
def __init__(self, value):
self.value = value
self.left_child = None
self.right_child = None
Here it is. Our binary tree
class.
When we instantiate an object, we pass the value
(the data of the node) as a parameter. Look at the left_child
and the right_child
. Both are set to None
.
Why?
Because when we create our node
, it doesn’t have any children. We just have the node data
.
Let’s test it:
tree = BinaryTree('a')
print(tree.value) # a
print(tree.left_child) # None
print(tree.right_child) # None
That’s it.
We can pass the string
‘a
’ as the value
to our Binary Tree node
. If we print the value
, left_child
, and right_child
, we can see the values.
Let’s go to the insertion part. What do we need to do here?
We will implement a method to insert a new node
to the right
and to the left
.
Here are the rules:
If the current
node
doesn’t have aleft child
, we just create a newnode
and set it to the current node’sleft_child
.If it does have the
left child
, we create a new node and put it in the currentleft child
’s place. Allocate thisleft child node
to the new node’sleft child
.
Let’s draw it out. :)
Here’s the code:
def insert_left(self, value):
if self.left_child == None:
self.left_child = BinaryTree(value)
else:
new_node = BinaryTree(value)
new_node.left_child = self.left_child
self.left_child = new_node
Again, if the current node doesn’t have a left child
, we just create a new node and set it to the current node’s left_child
. Or else we create a new node and put it in the current left child
’s place. Allocate this left child node
to the new node’s left child
.
And we do the same thing to insert a right child node
.
def insert_right(self, value):
if self.right_child == None:
self.right_child = BinaryTree(value)
else:
new_node = BinaryTree(value)
new_node.right_child = self.right_child
self.right_child = new_node
Done. :)
But not entirely. We still need to test it.
Let’s build the followingtree
:
To summarize the illustration of this tree:
a
node
will be theroot
of ourbinary Tree
a
left child
isb
node
a
right child
isc
node
b
right child
isd
node
(b
node
doesn’t have aleft child
)c
left child
ise
node
c
right child
isf
node
both
e
andf
nodes
do not have children
So here is the code for the tree
:
a_node = BinaryTree('a')
a_node.insert_left('b')
a_node.insert_right('c')
b_node = a_node.left_child
b_node.insert_right('d')
c_node = a_node.right_child
c_node.insert_left('e')
c_node.insert_right('f')
d_node = b_node.right_child
e_node = c_node.left_child
f_node = c_node.right_child
print(a_node.value) # a
print(b_node.value) # b
print(c_node.value) # c
print(d_node.value) # d
print(e_node.value) # e
print(f_node.value) # f
Insertion is done.
Now we have to think about tree
traversal.
We have two options here: DepthFirst Search (DFS) and BreadthFirst Search (BFS).
So let’s dive into each tree traversal type.
DepthFirst Search (DFS)
DFS explores a path all the way to a leaf before backtracking and exploring another path. Let’s take a look at an example with this type of traversal.
The result for this algorithm will be 1–2–3–4–5–6–7.
Why?
Let’s break it down.
Go to the
left child
(2). Print it.Then go to the
left child
(3). Print it. (Thisnode
doesn’t have any children)Backtrack and go the
right child
(4). Print it. (Thisnode
doesn’t have any children)Backtrack to the
root
node
and go to theright child
(5). Print it.Go to the
left child
(6). Print it. (Thisnode
doesn’t have any children)Backtrack and go to the
right child
(7). Print it. (Thisnode
doesn’t have any children)Done.
When we go deep to the leaf and backtrack, this is called DFS algorithm.
Now that we are familiar with this traversal algorithm, we will discuss types of DFS: preorder
, inorder
, and postorder
.
Preorder
This is exactly what we did in the above example.
def pre_order(self):
print(self.value)
if self.left_child:
self.left_child.pre_order()
if self.right_child:
self.right_child.pre_order()
Inorder
The result of the inorder algorithm for this tree
example is 3–2–4–1–6–5–7.
The left first, the middle second, and the right last.
Now let’s code it.
def in_order(self):
if self.left_child:
self.left_child.in_order()
print(self.value)
if self.right_child:
self.right_child.in_order()
Postorder
The result of the post order
algorithm for this tree
example is 3–4–2–6–7–5–1.
The left first, the right second, and the middle last.
Let’s code this.
def post_order(self):
if self.left_child:
self.left_child.post_order()
if self.right_child:
self.right_child.post_order()
print(self.value)
BreadthFirst Search (BFS)
BFS algorithm traverses the tree
level by level and depth by depth.
Here is an example that helps to better explain this algorithm:
So we traverse level by level. In this example, the result is 1–2–5–3–4–6–7.
Level/Depth 0: only
node
with value 1Level/Depth 1:
nodes
with values 2 and 5Level/Depth 2:
nodes
with values 3, 4, 6, and 7
Now let’s code it.
def bfs(self):
queue = Queue()
queue.put(self)
while not queue.empty():
current_node = queue.get()
print(current_node.value)
if current_node.left_child:
queue.put(current_node.left_child)
if current_node.right_child:
queue.put(current_node.right_child)
To implement a BFS algorithm, we use the queue
data structure to help.
How does it work?
Here’s the explanation.
Binary Search tree
“A Binary Search Tree is sometimes called ordered or sorted binary trees, and it keeps its values in sorted order, so that lookup and other operations can use the principle of binary search” — Wikipedia
An important property of a Binary Search Tree
is that the value of a Binary Search Tree
node
is larger than the value of the offspring of its left child
, but smaller than the value of the offspring of its right child.
”
Here is a breakdown of the above illustration:
**A **is inverted. The
subtree
7–5–8–6 needs to be on the right side, and thesubtree
2–1–3 needs to be on the left.**B **is the only correct option. It satisfies the
Binary Search Tree
property.**C **has one problem: the
node
with the value 4. It needs to be on the left side of theroot
because it is smaller than 5.
Let’s code a Binary Search Tree!
Now it’s time to code!
What will we see here? We will insert new nodes, search for a value, delete nodes, and the balance of the tree
.
Let’s start.
Insertion: adding new nodes to our tree
Imagine that we have an empty tree
and we want to add new nodes
with the following values in this order: 50, 76, 21, 4, 32, 100, 64, 52.
The first thing we need to know is if 50 is the root
of our tree.
We can now start inserting node
by node
.
76 is greater than 50, so insert 76 on the right side.
21 is smaller than 50, so insert 21 on the left side.
4 is smaller than 50.
Node
with value 50 has aleft child
21. Since 4 is smaller than 21, insert it on the left side of thisnode
.32 is smaller than 50.
Node
with value 50 has aleft child
21. Since 32 is greater than 21, insert 32 on the right side of thisnode
.100 is greater than 50.
Node
with value 50 has aright child
76. Since 100 is greater than 76, insert 100 on the right side of thisnode
.64 is greater than 50.
Node
with value 50 has aright child
76. Since 64 is smaller than 76, insert 64 on the left side of thisnode
.52 is greater than 50.
Node
with value 50 has aright child
76. Since 52 is smaller than 76,node
with value 76 has aleft child
64. 52 is smaller than 64, so insert 54 on the left side of thisnode
.
Do you notice a pattern here?
Let’s break it down.
Now let’s code it.
class BinarySearchTree:
def __init__(self, value):
self.value = value
self.left_child = None
self.right_child = None
def insert_node(self, value):
if value <= self.value and self.left_child:
self.left_child.insert_node(value)
elif value <= self.value:
self.left_child = BinarySearchTree(value)
elif value > self.value and self.right_child:
self.right_child.insert_node(value)
else:
self.right_child = BinarySearchTree(value)
It seems very simple.
The powerful part of this algorithm is the recursion part, which is on line 9 and line 13. Both lines of code call the insert_node
method, and use it for its left
and right
children
, respectively. Lines 11
and 15
are the ones that do the insertion for each child
.
Let’s search for the node value… Or not…
The algorithm that we will build now is about doing searches. For a given value (integer number), we will say if our Binary Search Tree
does or does not have that value.
An important item to note is how we defined the tree insertion algorithm. First we have our root
node
. All the left subtree
nodes
will have smaller values than the root
node
. And all the right subtree
nodes
will have values greater than the root
node
.
Let’s take a look at an example.
Imagine that we have this tree
.
Now we want to know if we have a node based on value 52.
Let’s break it down.
Now let’s code it.
class BinarySearchTree:
def __init__(self, value):
self.value = value
self.left_child = None
self.right_child = None
def find_node(self, value):
if value < self.value and self.left_child:
return self.left_child.find_node(value)
if value > self.value and self.right_child:
return self.right_child.find_node(value)
return value == self.value
Let’s beak down the code:
Lines 8 and 9 fall under rule #1.
Lines 10 and 11 fall under rule #2.
Line 13 falls under rule #3.
How do we test it?
Let’s create our Binary Search Tree
by initializing the root
node
with the value 15.
bst = BinarySearchTree(15)
And now we will insert many new nodes
.
bst.insert_node(10)
bst.insert_node(8)
bst.insert_node(12)
bst.insert_node(20)
bst.insert_node(17)
bst.insert_node(25)
bst.insert_node(19)
For each inserted node
, we will test if our find_node
method really works.
print(bst.find_node(15)) # True
print(bst.find_node(10)) # True
print(bst.find_node(8)) # True
print(bst.find_node(12)) # True
print(bst.find_node(20)) # True
print(bst.find_node(17)) # True
print(bst.find_node(25)) # True
print(bst.find_node(19)) # True
Yeah, it works for these given values! Let’s test for a value that doesn’t exist in our Binary Search Tree
.
print(bst.find_node(0)) # False
Oh yeah.
Our search is done.
Deletion: removing and organizing
Deletion is a more complex algorithm because we need to handle different cases. For a given value, we need to remove the node
with this value. Imagine the following scenarios for this node
: it has no children
, has a single child
, or has two children
.

Scenario #1: A
node
with nochildren
(leaf
node
).
# 50 50
# / \ / \
# 30 70 (DELETE 20) > 30 70
# / \ \
# 20 40 40
If the node
we want to delete has no children, we simply delete it. The algorithm doesn’t need to reorganize the tree
.

Scenario #2: A
node
with just one child (left
orright
child).
# 50 50
# / \ / \
# 30 70 (DELETE 30) > 20 70
# /
# 20
In this case, our algorithm needs to make the parent of the node
point to the child
node. If the node
is the left child
, we make the parent of the left child
point to the child
. If the node
is the right child
of its parent, we make the parent of the right child
point to the child
.

Scenario #3: A
node
with two children.
# 50 50
# / \ / \
# 30 70 (DELETE 30) > 40 70
# / \ /
# 20 40 20
When the node
has 2 children, we need to find the node
with the minimum value, starting from the node
’sright child
. We will put this node
with minimum value in the place of the node
we want to remove.
It’s time to code.
def remove_node(self, value, parent):
if value < self.value and self.left_child:
return self.left_child.remove_node(value, self)
elif value < self.value:
return False
elif value > self.value and self.right_child:
return self.right_child.remove_node(value, self)
elif value > self.value:
return False
else:
if self.left_child is None and self.right_child is None and self == parent.left_child:
parent.left_child = None
self.clear_node()
elif self.left_child is None and self.right_child is None and self == parent.right_child:
parent.right_child = None
self.clear_node()
elif self.left_child and self.right_child is None and self == parent.left_child:
parent.left_child = self.left_child
self.clear_node()
elif self.left_child and self.right_child is None and self == parent.right_child:
parent.right_child = self.left_child
self.clear_node()
elif self.right_child and self.left_child is None and self == parent.left_child:
parent.left_child = self.right_child
self.clear_node()
elif self.right_child and self.left_child is None and self == parent.right_child:
parent.right_child = self.right_child
self.clear_node()
else:
self.value = self.right_child.find_minimum_value()
self.right_child.remove_node(self.value, self)
return True
 To use the
clear_node
method: set theNone
value to all three attributes — (value
,left_child
, andright_child
)
def clear_node(self):
self.value = None
self.left_child = None
self.right_child = None
 To use the
find_minimum_value
method: go way down to the left. If we can’t find anymore nodes, we found the smallest one.
def find_minimum_value(self):
if self.left_child:
return self.left_child.find_minimum_value()
else:
return self.value
Now let’s test it.
We will use this tree
to test our remove_node
algorithm.
# 15
# / \
# 10 20
# / \ / \
# 8 12 17 25
# \
# 19
Let’s remove the node
with the value
8. It’s a node
with no child.
print(bst.remove_node(8, None)) # True
bst.pre_order_traversal()
# 15
# / \
# 10 20
# \ / \
# 12 17 25
# \
# 19
Now let’s remove the node
with the value
17. It’s a node
with just one child.
print(bst.remove_node(17, None)) # True
bst.pre_order_traversal()
# 15
# / \
# 10 20
# \ / \
# 12 19 25
Finally, we will remove a node
with two children. This is the root
of our tree
.
print(bst.remove_node(15, None)) # True
bst.pre_order_traversal()
# 19
# / \
# 10 20
# \ \
# 12 25
The tests are now done. :)
That’s all for now!
We learned a lot here.
Congrats on finishing this dense content. It’s really tough to understand a concept that we do not know. But you did it. :)
This is one more step forward in my journey to learning and mastering algorithms and data structures. You can see the documentation of my complete journey here on my Renaissance Developer publication.
Have fun, keep learning and coding.
Here are my Medium, Twitter, GitHub, and LinkedIn accounts.☺
Additional resources
How To Not Be Stumped By Trees by the talented Vaidehi Joshi
Binary Tree Implementation and Tests by TK
Coursera Course: Data Structures by University of California, San Diego
Coursera Course: Data Structures and Performance by University of California, San Diego
Binary Search Tree concepts and Implementation by Paul Programming
Oh, boy! Thanks for the detailed post. Must've been a ton of hard work. Is it okay if I copy/adapt this article for PHP and post it on my blog with proper credits? The PHP world sucks, in that we don't seem to care about computer science, and such posts will help a lot. 😇
Sure, share with me when you publish it
Absolutely! Thanks much.
But it's going to be an exact replica, just with PHP code. So don't pin many hopes on it. :P
Love the handdrawn images ♥