Tree Traversal in Data Structure Using Python [with Codes]

Data is getting generated at an exponential rate, with over 2.5 quintillion bytes created daily – this massive influx of information pressures organizations to adopt advanced techniques for efficiently managing and manipulating diverse datasets.

One such technique is tree traversal, which is essential for navigating and processing hierarchical data structures. They enable efficient data extraction, manipulation, and visualization across diverse applications in computing, database management, etc.

In this article, we will learn what tree traversal is in data structure by exploring various aspects of tree traversal, from its techniques and process to implementation and challenges. Let’s start by understanding what tree traversal is all about!

What is Tree in Data Structure?

To answer the question of what tree traversal is in data structure, you need first to understand tree data structure. 

A tree, by definition, is a data structure composed of nodes connected by edges, forming a hierarchy without cycles. The primary components include the root (top node), parent and child nodes, leaves (nodes without children), and internal nodes with at least one child.

There are numerous types of trees, the most common being binary trees, where each node has a maximum of two children, and binary search trees (BST), which organize nodes based on their values for efficient searching.

• Tree Traversal

Tree traversal in data structures is the method used to visit and process each node in a tree exactly once. Since trees follow a hierarchical rather than a linear structure, traversal ensures that every node is systematically accessed. This process is essential for various operations, such as searching, updating, or performing calculations on the nodes.

• Types of Tree Traversal

Tree traversal algorithms in data structure can be categorized into two main types: depth-first search (DFS) and breadth-first search (BFS).

• Depth-First Search (DFS): This tree traversal algorithm in data structure explores as far down a branch as possible before backtracking to visit other branches. DFS can be further divided into:

1. In-order Traversal: Visits the left subtree, root, and then the right subtree. Commonly used in binary search trees to retrieve nodes in ascending order.

2. Pre-order Traversal: Visits the root first, followed by the left and right subtrees. Useful for copying or reconstructing trees.

3. Post-order Traversal: Visits the left and right subtrees before the root. This approach is beneficial for deleting or freeing nodes.

• Breadth-First Search (BFS): Also known as level-order traversal, BFS visits all nodes at the current level before progressing to the next level. This ensures that nodes are processed layer by layer, making it effective for tasks like finding the shortest path or conducting level-based analyses.

• Significance of Tree Traversal

Tree traversal in data structure is crucial for efficiently managing data, as it enables structured access to hierarchical information. The algorithms help search and modify data and play a vital role in applications like parsing XML/HTML documents, organizing file systems, and creating decision trees in AI.

Let’s now focus on the various data structures that can be used to implement tree traversal.

Data Structures Used to Implement Tree Traversal

In tree traversal in the data structure, iterating over each node systematically is essential to ensure every node is visited once. Since trees are hierarchical and non-linear, traversing them requires temporarily storing nodes for later visits. We can address this requirement by using two data structures: stacks and queues.

1) Stack Data Structure

A stack follows the Last In, First Out (LIFO) principle, where the most recently added element is removed first. This property makes stacks ideal for depth-first search (DFS) in tree traversal.

DFS explores nodes deeply before backtracking, efficiently managing nodes by using a stack. It pushes new nodes onto the stack as it explores branches and pops them when it backtracks.

– Operations in Stack:

• • Push: Insert an element at the top.
  • Pop: Removes and returns the top element.
  • Peek: Retrieves the top element without removing it.
  • IsEmpty: Checks if the stack is empty.

– Use Cases in Tree Traversal:

• • Manages function calls in recursive algorithms.
  • Expression parsing and evaluation.
  • DFS traversal in trees and graphs.

2) Queue Data Structure

Unlike stack, a queue adheres to the First In, First Out (FIFO) principle, where the earliest added element is the first to be removed. This behavior aligns with breadth-first search (BFS), which explores all nodes at the current level before progressing to the next.

A queue manages this by enqueuing child nodes and dequeuing them in the order they were added. Thus, the queue data structure is used for BFS.

– Operations in Queue:

• • Enqueue: Adds an element to the Rear.
  • Dequeue: Removes and returns the front element.
  • Front: Retrieves the front element without removing it.
  • IsEmpty: Checks if the queue is empty.

– Use Cases in Tree Traversal:

• • Level-order traversal of trees.
  • Task scheduling and process management.
  • BFS traversal for exploring graphs.

Both stacks and queues are fundamental to tree traversal as they ensure the nodes are systematically accessed based on traversal strategies. Their structured approach simplifies complex tree operations. 

It’s time to dive deep into the process used by the different tree traversal techniques. Only after you fully understand the process can you proceed toward using and implementing them.

Tree Traversal Techniques – Process & Uses

As mentioned earlier, tree traversal techniques are essential for systematically visiting each node in a tree, with two primary methods, DFS and BFS, offering distinct advantages. While DFS excels at exploring tree depth, making it ideal for tasks like expression evaluation and pathfinding, BFS, on the other hand, prioritizes breadth, visiting nodes level by level, which is critical for shortest-path calculations and resource distribution. 

You must understand the unique strengths of each technique as it can enable you to apply them in diverse computational scenarios, allowing you to ensure efficient data handling and problem-solving.

Below, we will look at the different techniques available under DFS and BFS and explore the processes they follow.

1) Depth-First Search (DFS)

Depth-first search delves into the tree by exploring as far down a branch as possible before retracing steps and venturing down the next. If you want to understand with an analogy, imagine this approach like peeling layers of an onion where it steadily descends to the deepest point before resurfacing.

There are three principal forms of DFS viz. Inorder, Preorder, and Postorder. These forms differ in the sequence in which they handle root, left, and right nodes. Let’s understand them one by one.

• Inorder Traversal

Inorder traversal glides through the tree by exploring the left subtree, visiting the root, and finally shifting to the right. This pattern is potent in binary search trees (BST), as it naturally retrieves nodes in ascending order.

• Algorithm:

Check if the node exists, i.e., assess whether the current node is not null.

-> Recurse on the left subtree 

-> Process the current node

-> Recurse on the right subtree

• Use Cases:

-> Binary Search Trees (BST): Inorder traversal lists nodes in sorted order, making it crucial for searching or validating BST properties.

-> Expression Trees: Derive infix expressions, where operators appear between operands.

-> Syntax Analysis: Inorder traversal plays a role in compiler design and parsing.

• Time Complexity: O(N)
• Space Complexity: O(h) (where h is the tree height)

• Preorder Traversal (Root -> Left -> Right)

Pre-order traversal resembles the blueprint for constructing or reconstructing a tree and follows a ‘root-left-right’ policy. This is because it visits the root before its subtrees.

• Algorithm:

Check if the node exists, i.e., assess whether the current node is not null.

-> Process the current node

-> Recurse on the left subtree

-> Recurse on the right subtree

• Use Cases:

-> Tree Duplication: Pre-order traversal is employed to create a copy of an entire tree structure.

-> Prefix Expressions: Expression trees rely on pre-order traversal to derive prefix notation.

-> Pathfinding: Pre-order traversal can trace all paths from the root to the leaf in scenarios like network routing or AI decision trees.

• Time Complexity: O(N)
• Space Complexity: O(h)

• Postorder Traversal (Left -> Right -> Root)

This method requires patience, as it fully explores the left and right subtrees before focusing on the root. Developers frequently use it for tasks like deleting or freeing memory, ensuring nodes are processed from the ground up.

• Algorithm:

Check if the node exists, i.e., assess whether the current node is not null.

-> Recurse on the left subtree (process all nodes on the left side first).

-> Recurse on the right subtree (move to the right after the left is complete).

-> Process the current node (visit the node only after its children are processed).

• Use Cases:

-> Memory Management: Post-order traversal is used to delete trees and release memory.

-> Expression Trees: Generates postfix expressions by visiting operands before operators.

-> Garbage Collection: Post-order traversal underpins algorithms for deallocating objects in memory management systems.

• Time Complexity: O(N)
• Space Complexity: O(h)

2) Breadth-First Search (BFS)

Unlike DFS, which dives deep, BFS, on the contrary, spreads out. It systematically visits nodes level by level. If we use an analogy, it’s like scanning each floor of a building before proceeding to the next.

This method ensures that no node at a given depth is overlooked, making it critical for operations involving broad and horizontal sweeps. The principal form of BFS is Level Order Traversal.

• Level Order Traversal (Left to Right, Level by Level)

Level order traversal exemplifies the essence of BFS as it prioritizes breadth by visiting each node at a level before descending further.

• Algorithm:

Initialize an empty queue and enqueue the root node (start by adding the root to the queue).

While the queue is not empty (as long as nodes remain to be visited):

-> Dequeue the front node (remove the first node from the queue and process it).

-> Enqueue the left child if it exists (add the left child to the queue for future processing).

-> Enqueue the right child if it exists (add the right child after the left one is enqueued).

• Use Cases:

-> Tree Serialization and Deserialization: Used to encode trees for transmission and reconstruct them on the receiving end.

-> Shortest Path Algorithms: Plays a critical role in identifying the shortest path in unweighted graphs, which is critical in networking and game development.

-> Resource Allocation: Helps distribute workloads evenly across servers or nodes.

• Time Complexity: O(N)
• Space Complexity: O(N) (due to queue usage)

You must remember that tree traversal techniques are more than mere algorithms because they are the essential tools for revealing hierarchical data’s hidden structure and potential. If you deal with data, you must master these methods as they pave the way for efficient data management and provide robust solutions for complex computational problems.

After covering all the major aspects of the processes of tree traversal, we will understand the implementation of these various tree traversal techniques.

Steps to Implement Tree Traversal Techniques 

In this section, we will implement a binary tree traversal in data structure. We will be using various tree traversal techniques in Python and visually analyze each step of the process.

• General Steps

I’ll start by providing you with the generic steps that need to be performed for binary tree traversal in data structure using different techniques.

Step 1: Creating a basic tree structure

The first step involves creating a class to define a simple structure for the binary tree for each tree node. Each node has a val (value) and pointers to its left and right child nodes.

# creating a class for defining a simple structure for each tree node
class TreeNode:
def __init__(self, val):
self.val = val
self.left = None
self.right = None

Step 2: Creating Traversal Function

The traversal function is the primary function that defines how traversal is performed. Later, we will understand this step in detail for different techniques. But before that, let’s understand the general idea behind this function.

It is designed to systematically visit and process each node in a tree, ensuring all nodes are accessed in a defined sequence. It begins by initializing an empty list answer to store the nodes in the order they are visited. The core traversal logic is managed through a nested function or iterative process, which directs how nodes are explored and how their children are handled.

Based on the traversal technique (in-order, pre-order, post-order, or level-order), the function visits nodes in a specific pattern, iteratively traversing subtrees or progressing through levels. Upon completion, the list answer stores the traversal results.

The generic code to create this function will be

# creating traversal function
def Traversal(root):
answer = [] def traverse(root):
if not root:
return

# code for performing traversal on a tree which will depend on the technique used as per
pass  # to be replaced with technique-specific logic
traverse(root)
return answer

Note – The <NameofTechnique> can be inorder, pre-order, postorder, and levelorder.

Step 3: Constructing Binary Tree

The binary tree is now constructed with a root node and child nodes. We will create a binary tree that looks like this-

Following is the code for constructing the binary tree.

# constructing a binary tree
root = TreeNode(1)
root.left = TreeNode(2)
root.right = TreeNode(3)
root.left.left = TreeNode(4)
root.left.right = TreeNode(5)
root.right.left = TreeNode(6)
root.right.right = TreeNode(7)

Step 4: Calling Traversal Function

In the last step, we will call the traversal function to print the results. The generic code for this would be-

# printing result of traversal
print(" Traversal Result Result:", Traversal(root))

In the following sections, we will apply this traversal function, present the results, and visually break down each step leading to the final output.

• In-Order Traversal Steps

To perform binary search tree traversal in data structure, you need to start with creating the class TreeNode, which has the structure for the binary tree. Once done, create the traversal function.

The in-order traversal function performs traversal by first recursively traversing the left subtree, then visiting the current node by appending its value to answer, and finally traversing the right subtree. After traversal, the answer contains the nodes in ascending order for binary search trees (BST).

def inorderTraversal(root):
answer = [] def traverse(root):
if not root:
return
traverse(root.left)
answer.append(root.val)
traverse(root.right)
traverse(root)
return answer

The next step involves constructing the binary tree. Once that is done, you can call the traversal function.

# printing result of in-order traversal
print("In-order Traversal Result:", inorderTraversal(root))

 In-order Traversal Result : [ 4 , 2 , 5 , 1 , 6 , 3 , 7 ] 

Let’s visually understand the step-by-step process followed by in-order binary search tree traversal in the data structure to achieve the result shown above.

Step 1: Visits Node 4 (Leftmost Node)

Starts at the root (Node 1) but recursively traverses the left subtree. Moves left to Node 2, then continues left to Node 4. Node 4 is the leftmost node, so it is visited first.

Step 2: Visits Node 2 (Parent of 4 and 5)

After visiting Node 4, backtracks to its parent, Node 2. 

Step 3: Visits Node 5 (Right Child of Node 2)

From Node 2, moves to the right subtree and visits Node 5.

Step 4: Visits Node 1 (Root Node)

After completing the left subtree of Node 1, visits the root (Node 1).

Step 5: Visits Node 6 (Left Child of Node 3)

Now traverses the right subtree of Node 1. Moves to Node 3 and starts by visiting its left child, Node 6.

Step 6: Visits Node 3 (Parent of 6 and 7)

After visiting Node 6, backtracks to its parent, Node 3.

Step 7: Visits Node 7 (Right Child of Node 3)

Finally, visits the right child of Node 3, which is Node 7.

• Pre-Order Traversal Steps

The first step is to create the class TreeNode. Once done, create the in-order traversal function where traversal is performed by first visiting the current node and appending its value to answer, then recursively traversing the left subtree, followed by the right subtree. After traversal, the answer contains the nodes in the order they were first encountered.

def preorderTraversal(root):
answer = [] def traverse(root):
if not root:
return
answer.append(root.val)
traverse(root.left)
traverse(root.right)
traverse(root)
return answer

Now, you need to construct the binary tree, followed by calling it the traversal function.

# printing result of pre-order traversal
print("Pre-order Traversal Result:", preorderTraversal(root))

 Pre-order Traversal Result : [ 1 , 2 , 4 , 5 , 3 , 6 , 7 ] 

We’ll now visually explain the steps followed by the traversal process.

Step 1: Visits Node 1 (Root Node)

Starts at the root (Node 1) and visit it first. 

Step 2: Visits Node 2 (Left Child of Node 1)

Moves to the left subtree and visits Node 2.

Step 3: Visits Node 4 (Left Child of Node 2)

Continues traversing the left subtree by visiting Node 4. 

Step 4: Visits Node 5 (Right Child of Node 2)

After visiting the left subtree, visits Node 5.

Step 5: Visits Node 3 (Right Child of Node 1)

Now, the pre-order traversal function traverses the right subtree of Node 1 and visits Node 3.

Step 6: Visits Node 6 (Left Child of Node 3)

Visits Node 6, the left child of Node 3.

Step 7: Visits Node 7 (Right Child of Node 3)

Finally, visit the right child of Node 3, Node 7.

• Post-Order Traversal Steps

Start by creating the class TreeNode, followed by creating the post-order traversal function. The post-order traversal function performs traversal by first recursively traversing the left subtree, then the right subtree, and finally visiting the current node by appending its value to the answer.

def postorderTraversal(root):
answer = [] def traverse(root):
if not root:
return
traverse(root.left)
traverse(root.right)
answer.append(root.val)
traverse(root)
return answer

After constructing the binary tree, call the traversal function.

# printing result of post-order traversal
print("Post-order Traversal Result:", postorderTraversal(root))

 Post-order Traversal Result : [ 4 , 5 , 2 , 6 , 7 , 3 , 1 ] 

Let’s visually understand how the post-order traversal process works in the traversal function.

Step 1: Visits Node 4 (Leftmost Leaf Node)

Starts at the root (Node 1) and recursively traverses the left subtree to visit Node 4.

Step 2: Visits Node 5 (Right Child of Node 2)

After Node 4, the function traverses the right child of Node 2, visiting Node 5. 

Step 3: Visits Node 2 (Parent of 4 and 5)

Once the left and right children of Node 2 are visited, the function visits Node 2.

Step 4: Visits Node 6 (Left Child of Node 3)

Moves to the right subtree of Node 1 and visits Node 6.

Step 5: Visits Node 7 (Right Child of Node 3)

After visiting Node 6, traverses and visits Node 7.

Step 6: Visits Node 3 (Parent of 6 and 7)

Visits Node 3 after visiting its children. 

Step 7: Visits Node 1 (Root Node)

Finally, visits the root node (Node 1) after completing both subtrees.

• Level-Order Traversal Steps

Create the class TreeNode. Once completed, create the traversal function. The level-order traversal function performs traversal by visiting nodes level by level, starting from the root. A queue is used to manage nodes, where each node is dequeued, its value is appended to an answer, and its children are enqueued.

from collections import deque
def levelOrderTraversal(root):
answer = [] if not root:
return answer
queue = deque([root])
while queue:
node = queue.popleft()
answer.append(node.val)
if node.left:
queue.append(node.left)
if node.right:
queue.append(node.right)
return answer

Construct the binary tree and proceed to call the level-order traversal function.

# printing result of level-order traversal
print("Level-order Traversal Result:", levelOrderTraversal(root))

 Level-order Traversal Result : [ 1 , 2 , 3 , 4 , 5 , 6 , 7 ] 

Let’s now have a visual understanding of how the level-order traversal functions.

Step 1: Visits Node 1 (Root Node)

Starts at the root (Node 1) and visits it first.

Step 2: Visits Nodes 2 and 3 (Level 1)

Moves to the next level and visits Node 2 (left) and Node 3 (right).

Step 3: Visits Nodes 4, 5, 6, and 7 (Level 2)

Visits the next level and traverses Nodes 4, 5, 6, and 7.

As you can see, the traversal function is critical as it dictates the order in which the nodes in the tree are visited. Now, let’s have a brief look at a few more traversal techniques.

Additional Tree Traversal Techniques

Tree traversal extends beyond basic in-order, pre-order, and post-order techniques. Specialized methods like boundary traversal, diagonal traversal, and zigzag traversal offer unique ways to navigate and extract insights from binary trees.

1) Boundary Traversal

Boundary traversal focuses on visiting the outermost nodes of a binary tree. This includes nodes from the left boundary, leaf nodes, and the right boundary in a specific order. The traversal provides a holistic view of the tree’s periphery.

Algorithm for Boundary Traversal:

1. Start at the root node.
2. Traverse the left boundary, excluding the leftmost leaf.
3. Visit all leaf nodes from left to right.
4. Traverse the right boundary, excluding the rightmost leaf.

Example Traversal:

Boundary Traversal: 13 🡪 11 🡪 9 🡪 14 🡪 16 🡪 17 🡪 15

Applications:

• Visualizes the outer structure of trees, making it easier to identify the overall shape and detect imbalances or asymmetries in the tree’s growth.
• Aids in operations like pruning or modifying boundary nodes.

2) Diagonal Traversal

Diagonal traversal involves visiting nodes diagonally from top-left to bottom-right. This method blends depth-first and breadth-first approaches, capturing nodes along diagonals.

Algorithm for Diagonal Traversal:

1. Start at the root node.
2. Visit the current node and its right child.
3. Proceed to the left child of the node.
4. Repeat this process until no nodes remain on the diagonal, then move to the next diagonal.

Example Traversal:

1) Diagonal 0: 13 15 17

2) Diagonal 1: 11 12 16

3) Diagonal 2: 9 14

Applications:

• It groups nodes along the same diagonal, thereby providing a diagonal perspective of the tree. Such a perspective helps highlight structural relationships based on depth and proximity that may not be visible in standard traversals.
• It helps partition nodes along diagonals in data processing, mirroring real-world hierarchical relationships. This aids in identifying data layers that align by depth and separation, enhancing clustering and processing.

3) Zigzag Traversal

Zigzag traversal alternates between left-to-right and right-to-left directions at each level of the tree. This approach captures the tree’s structure dynamically.

Algorithm for Zigzag Traversal:

1. Use two stacks to manage nodes at the current and next levels.
2. Start by pushing the root into the current level stack.
3. Pop nodes from the current stack and push children into the next stack in left-to-right or right-to-left order.
4. Swap the stacks after each level and toggle the traversal direction.

Example Traversal:

Zigzag Traversal: 4 🡪 5 🡪 2 🡪 1 🡪 3

Applications:

• Simulates hierarchical zigzag patterns in data, which helps highlight alternating patterns often used in tree-like datasets to reflect layered or nested structures.
• Alternating-level processing simulates real-world hierarchical systems (e.g., organizational charts or decision trees), where alternating directions at each level enhance exploration, balance load distribution, and uncover patterns not visible with linear traversals. Zigzag traversal shifts the traversal direction at each level, enabling the system to process data from multiple perspectives and fully explore hierarchical layers.

By going beyond the conventional techniques, these traversal methods can help you extend the capabilities of tree operations, allowing you to perform advanced data extraction and manipulation in binary trees.

After discussing the applications of various tree traversal techniques, let us now see how do we use tree traversal in the real world.

Application of Tree Traversal in Data Structure

Tree traversal in data structure has widespread applications across various fields. In computer science, they are crucial for evaluating expressions, building syntax trees in compilers, and parsing hierarchical data like XML and HTML to generate Document Object Models (DOM).

In databases, traversing B-trees and B+ trees facilitates efficient indexing, while in machine learning, decision trees are navigated for predictions and insights.

Additionally, tree traversal aids in analyzing game strategies, routing network paths, and rendering complex graphics. For binary trees, specific traversal orders like in-order yield sorted elements in binary search trees, and level-order traversal implements breadth-first search for hierarchical exploration. Beyond technology, applications include genealogy tree exploration and representing organizational structures effectively.

To effectively use the various tree traversal techniques discussed so far, it is critical to know the various challenges associated with them. Below, we will explore the key challenges.

Challenges with Tree Traversal Techniques

There are many challenges with using tree traversal techniques, the most critical one being the following.

1) Handling Large Trees

Traversing large or deeply nested trees can result in significant time and space complexity. This can lead to performance bottlenecks, especially if the traversal algorithm is inefficient.

2) Stack Overflow in Recursion

Recursive tree traversals can lead to stack overflow errors for deeply nested trees, as each recursive call adds to the call stack. This is a significant limitation of in-depth-first approaches.

3) Memory Usage in Iterative Approaches

Iterative methods often require additional data structures (like stacks or queues) to manage nodes at different levels, causing a spike in memory consumption.

4) Balancing Efficiency and Simplicity

Some traversal techniques, such as boundary or zigzag traversal, require complex logic and careful management of data structures. This can make the implementation more error-prone and harder to debug.

5) Unbalanced Trees

Traversal can become inefficient, especially in unbalanced trees. For example, an unbalanced binary tree can degrade to a linear structure, making the traversal as slow as O(n), which diminishes the benefits of hierarchical data organization.

6) Order-Specific Traversals

Traversals that depend on specific orders (e.g., diagonal or boundary traversals) require customized algorithms that are not as straightforward as the basic ones (in-order, pre-order, or post-order traversals).

7) Real-time Constraints

In applications that require real-time data processing (e.g., graphics rendering or AI decision trees), developers must optimize traversal to minimize delays, which poses challenges for large datasets.

Conclusion

Tree traversal is essential for efficiently managing hierarchical data structures. Several traversal techniques are available today, each offering unique benefits. While in-order traversal is key for extracting sorted data from binary search trees, pre-order helps with tree construction and copying.

Post-order is vital for deleting nodes and memory management. Level-order traversal (BFS) explores nodes level by level, which is useful for organizational structures. Apart from the fundamental ones, additional specialized techniques like boundary, zigzag, and diagonal traversal also exist that can help you reveal hidden patterns and external structures.

Despite the various challenges associated with them, like large data sets and unbalanced trees, these techniques continue to drive advancements in computing, AI, and data science.

FAQs

• What is the difference between Depth-First and Breadth-First Traversal?

DFS explores each branch as deeply as possible before backtracking, while BFS visits all nodes at the current level before moving to the next.

• Which traversal method is best for binary search trees (BST)?

In-order traversal is best for BSTs as it retrieves nodes in ascending order with O(n) time complexity and O(h) space, making it efficient for sorted data extraction and range queries.

• Can tree traversal be used on non-binary trees?

Yes, traversal techniques can be adapted for non-binary trees. You can do this by iterating over all child nodes at each level. The traversal techniques like pre-order, post-order, and level-order methods can handle multiple children by recursively or iteratively processing each child in sequence.

• Can tree traversal be parallelized?

Yes, you can do this by processing subtrees independently, particularly in BFS.

• What is the time complexity of tree traversal algorithms?

The time complexity for most tree traversal algorithms is O(n). Here, n refers to the number of nodes( in the tree).