Data structures serve as the backbone of efficient data organization and management within computer systems. They play a pivotal role in computer algorithms and software development, contributing significantly to designing streamlined and effective programs. Data structures are indispensable across various domains in computer science, ranging from Artificial Intelligence to Operating Systems. 

One of the most widely used non-linear data structures is trees. Trees portray a hierarchical arrangement, rendering ordering information irrelevant, unlike their linear counterparts, such as arrays, stacks, queues, and linked lists. 

Comprising nodes and pointers, a tree’s structure involves parent nodes with left and right children, presenting a versatile and fundamental concept in data organization. This brings us to the topic we will discuss extensively in this blog – binary tree representation in data structure.

Amongst different types of data structures are binary trees that come with more uses than most of the other types. Their most notable applications include peer-to-peer programming, search, cryptography, network routers with higher bandwidth than others, and 3D video games. We will now discuss in detail what binary trees in data science are, what are their types, and how are they represented. 

What are binary trees?

If you have worked on normal trees before or even know about their basics, you would know that there are no restrictions when it comes to the number of children that different nodes are allowed to have in these trees. Binary trees are a little different in this sense. Every parent or node in binary trees can have a maximum of only two children. 

In binary tree representation, this limitation, however, simplifies the structure, making binary trees highly efficient in scenarios where quick data retrieval is essential. The binary nature allows for straightforward decision-making processes commonly used in searching algorithms. Despite the restriction on the number of children, the versatility and speed of binary trees make them crucial elements in various computer science applications.

All nodes in a binary tree have three primary components – 

• a data element 

This component has the actual information or value held by the node. It could represent various types of data, such as numbers, strings, or other relevant content, depending on the context of the tree.

• a right reference 

The right reference is a pointer or reference that directs to the node’s right child. The right child is a node positioned to the right of the current node within the binary tree hierarchy. This reference helps traverse and explore the tree’s right branch.

• a left reference 

Similar to the right reference, the left reference is a pointer or reference that points to the node’s left child. The left child is positioned to the left of the current node, forming the left branch of the binary tree. This reference is important for navigating and analyzing the tree’s left subtree.

In combination, these three components create a cohesive structure, allowing for efficient organization, storage, and data retrieval within the binary tree representation. The data element holds the pertinent information, while the right and left references guide the relationships between nodes, forming the foundation for the tree’s hierarchical arrangement.

The node that lies at the top of the tree is referred to as the root node. Parent nodes are those that have children. Children nodes and parent nodes are connected to each other through references. Nodes that don’t have any children are referred to as leaf nodes.

It is clearly evident that nodes in binary trees can have one child, two children, or no children at all. Binary trees aren’t linear data structures like queues, arrays, stacks, and linked lists. They are hierarchical data structures instead. 

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Important properties of nodes in binary trees

A better understanding of these properties will help you in making the most of this discussion on binary trees. The depth of different nodes is defined as the number of nodes that exist on the way that connects the root to a particular node. That is why the depth of the root node is 0. On the other hand, the height of different nodes in a binary tree is the number of nodes that lie in the path that connects a particular node with the root node. That is why the height of leaf nodes is 0. 

As you can clearly see, the depth of a node is measured by starting from the root node and then going down to reach that node. On the other hand, when it comes to calculating the height, we start at the node in question and then journey towards the root node. Both the times, we start at 0. There are people who also measure height and depth from1 and not from 0, which isn’t wrong and is just what different people prefer. 

Now the maximum depth of a node is defined as the depth of a binary tree. Similarly, the maximum height of a node is defined as the height of a binary tree. So the height and depth of a binary tree are always the same. 

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What is a binary search tree? 

A binary search tree is the most common of all the other types of binary trees. It is a specialized binary tree that comes with properties that are different and more useful than any other form of a binary tree. What exactly is a binary search tree or BST? Just as its name suggests, a binary search tree is used to search data in the tree.

A BST comes with properties that allow it to facilitate efficient searches. A BST is a binary tree that has the key of the node that is smaller and greater than nodes in the right sub-tree and nodes in the left sub-tree respectively. 

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Advantages of binary search tree

There are several advantages of binary search tree that contribute to their widespread use in computer science and algorithm design:

• Efficient search operations: The binary search tree structure enables quick and efficient search operations. The binary nature of the tree ensures that at each node, the left subtree contains values smaller than the node, and the right subtree contains values greater. This property allows for a binary search algorithm, reducing the search space at each step.
• Simple insertion and deletion: Inserting and deleting elements in a binary search tree is quite straightforward. The binary structure allows for easy maintenance of the order, and the tree can be rebalanced if needed to maintain optimal search performance.
• Ordered data retrieval: In-order traversal of a binary search tree results in sorted data retrieval. This property is valuable in scenarios where data needs to be accessed or processed in sorted order without additional sorting operations.
• Space efficiency: Compared to other data structures like arrays, binary search trees are more space-efficient, especially when dealing with dynamic datasets. Nodes are allocated as needed, and memory is not pre-allocated, making it flexible for varying data sizes.
• Versatility in applications: Binary search trees find applications in various domains, including databases, symbol tables, and compilers. Their efficient search and ordered retrieval properties make them versatile where these operations are critical.
• Balanced binary search trees: When balanced, binary search trees ensure optimal performance in terms of search, insertion, and deletion operations. Balanced trees, such as AVL trees or Red-Black trees, maintain a balanced structure, preventing the tree from degenerating into a linked list and ensuring logarithmic time complexity for operations.

Representation of binary trees

Binary tree representation in data structure can be carried out in different ways, each with distinct pros and cons. The prominent methods are binary tree representation using linked list (Linked representation), storage representation of binary tree (Sequential representation), and linear representation.

1. Linked representation

Binary trees in linked representation are stored in the memory as linked lists. These lists have nodes that aren’t stored at adjacent or neighboring memory locations and are linked to each other through the parent-child relationship associated with trees. 

In this representation, each node has three different parts – 

Pointer that points towards the right node

This is the more common representation. All binary trees consist of a root pointer that points in the direction of the root node. When you see a root node pointing towards null or 0, you should know that you are dealing with an empty binary tree. The right and left pointers store the address of the right and left children of the tree. 

Pointer that points towards the left node 

The second part is another pointer pointing towards the left child node. Similar to the right pointer, this element establishes the connection between the current node and its left child, contributing to the hierarchical structure of the binary tree.

Data element

The third part is the actual data element or value associated with the node. Depending on the context of the binary tree’s application, this could be any type of data, such as a number, a string, or another relevant piece of information.

This is the more common representation. All binary trees consist of a root pointer that points in the direction of the root node. When you see a root node pointing towards null or 0, you should know that you are dealing with an empty binary tree. The right and left pointers store the address of the right and left children of the tree. 

The linked representation of binary tree in data structure offers dynamic memory allocation, making it adaptable to varying tree sizes. The linked representation of binary tree in memory offers flexibility and ease of manipulation, making it a popular choice in applications requiring dynamic data structures.

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2. Sequential representation

Although it is simpler than linked representation, its inefficiency makes it a less preferred binary tree representation of the two. The inefficiency lies in the amount of space it requires for the storage of different tree elements. The sequential representation uses an array for the storage of tree elements.

The number of nodes a binary tree has defines the size of the array being used. The root node of the binary tree lies at the array’s first index. The index at which a particular node is stored will define the indices at which the right and left children of the node will be stored. An empty tree has null or 0 as its first index. 

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Despite its inefficiency, the sequential representation of binary tree in data structure is beneficial where memory allocation is a concern, as it uses a contiguous memory block. However, this memory representation of binary tree may become impractical for large or dynamic datasets due to its fixed array size.

3. Linear Representation of Binary Tree

Amongst the various representations of binary trees the linear representation focuses on organizing its elements linearly or sequentially, often for ease of traversal or processing. Two common linear ways of representing binary tree in memory are array-based linearization and in-order linearization.

Array-Based Linearization:

In the linear representation of binary tree, the tree is linearized into an array by traversing it in a specific order, such as level order or inorder. The elements are placed in the array based on their order in the traversal, allowing for a linear structure that simplifies access and manipulation. This method is particularly useful when a sequential representation is required for efficient storage and retrieval.

In-order Linearization:

In in-order linearization, the binary tree is traversed in an in-order fashion, meaning left subtree, root, and right subtree. This linearization results in a sequence of elements that follows the sorted order of the binary search tree. In-order linearization is often employed when the goal is to process or display elements in a sorted manner.

Types of binary trees

1. Full binary trees: Full binary trees are those binary trees whose nodes either have two children or none. In other words, a binary tree becomes a full binary tree when apart from leaves, all its other nodes have two children.Understanding the nature of full binary trees is fundamental, as their balanced structure enhances the efficiency of various tree operations.
2. Complete binary trees: Complete binary trees are those that have all their different levels completely filled. The only exception to this could be their last level, whose keys are predominantly on the left. A binary heap is often taken as an example of a complete binary tree.Complete binary trees, exemplified by binary heaps, showcase specific flexibility in their structure, optimizing memory use while facilitating efficient data retrieval.
3. Perfect binary trees: Perfect binary trees are binary trees whose leaves are present at the same level and whose internal nodes carry two children. A common example of a perfect binary tree is an ancestral family tree.With their balanced and symmetrical layout, perfect binary trees provide an idealized model for understanding the structural efficiency of certain tree-based data structures.
4. Pathological degenerate binary trees: Degenerate trees are those binary trees whose internal nodes have one child. Their performance levels are similar to linked lists. Learn more about the types of binary tree.Pathological degenerate binary trees serve as a reminder that not all tree structures are inherently advantageous, emphasizing the importance of selecting the appropriate type based on specific use cases.

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Benefits of binary trees

There are a myriad of advantages of binary tree that make them a valuable choice for storing and organizing data hierarchically. 

Some of the major advantages of binary tree include:

• An ideal way to go with the hierarchical way of storing data. 

The parent-child relationships among nodes create a natural hierarchy, facilitating efficient organization and retrieval of information.

• Reflect structural relationships that exist in the given data set. 

The arrangement of nodes, with each node having at most two children, captures relationships and dependencies within the data, enhancing the representation’s meaningfulness.

• Make insertion and deletion faster than linked lists and arrays.

The hierarchical structure allows for streamlined updates, making it easy to add or remove elements while maintaining the overall organization of the tree.

• A flexible way of holding and moving data.

The dynamic nature of the structure, with nodes being added or removed as needed, enables adaptability to changing data requirements. This flexibility is particularly advantageous in dynamic or evolving datasets.

• Are used to store as many nodes as possible.

The hierarchical organization ensures that the tree can accommodate a substantial volume of data while maintaining a balanced structure.

• Are faster than linked lists and slower than arrays when comes to accessing elements.

In terms of access speed, binary trees strike a balance between linked lists and arrays. While they may not match the swift access times of arrays, they outperform linked lists in speed. This balanced performance makes binary trees the preferred choice where a compromise between insertion/deletion efficiency and access speed is crucial.

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Conclusion

In this blog, we have discussed what binary trees in data structures are as well as talked about their types, their representations, and their benefits. The two major uses of the trees are for searching and storing data, and hence they are integral to the study of Data Science and its related fields.  

Understanding the different types of binary trees helps us handle various data organization challenges, while the ways we represent them offer insights into storing and retrieving information dynamically. Recognizing the inherent benefits of binary trees, we see them not just as computer science structures but as crucial elements supporting the foundations of Data Science.

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