5 Types of Binary Tree Explained [With Illustrations]

In computer science, various data structures help in arranging data in different forms. Among them, trees are widely used abstract data structures that simulate a hierarchical tree structure. A tree usually has a root value and subtrees that are formed by the child nodes from its parent nodes. Trees are non-linear data structures.

A general tree data structure has no limitation on the number of child nodes it can hold. Yet, this is not the case with a binary tree. This article will learn about a specific tree data structure – binary tree and types of binary tree.

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What is Binary Tree Data Structure?

A binary tree is a tree-type non-linear data structure with a maximum of two children for each parent. Every node in a binary tree has a left and right reference along with the data element. The node at the top of the hierarchy of a tree is called the root node. The nodes that hold other sub-nodes are the parent nodes.

A parent node has two child nodes: the left child and right child. Hashing, routing data for network traffic, data compression, preparing binary heaps, and binary search trees are some of the applications that use a binary tree.

Terminologies associated with Binary Trees and Types of Binary Trees

• Node: It represents a termination point in a tree.
• Root: A tree’s topmost node. 
• Parent: Each node (apart from the root) in a tree that has at least one sub-node of its own is called a parent node.
• Child: A node that straightway came from a parent node when moving away from the root is the child node.
• Leaf Node: These are external nodes. They are the nodes that have no child.
• Internal Node: As the name suggests, these are inner nodes with at least one child.
• Depth of a Tree: The number of edges from the tree’s node to the root is.
• Height of a Tree: It is the number of edges from the node to the deepest leaf. The tree height is also considered the root height.

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As you are now familiar with the terminologies associated with the binary tree and types of binary tree, it is time to understand the binary tree components. Check out our data science courses to learn in-depth about binary structure and components. 

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Understanding Properties of Binary Tree Or What Is Binary Tree?

At every level of it, the maximum number allowed for nodes stands at 2i.

The height of a binary tree stands defined as the longest path emanating from a root node to the tree’s leaf node.

What Is Binary Tree– More Than The Binary Tree Definition

Say a binary tree placed at a height equal to 3. In that case, the highest number of nodes for this height 3 stands equal to 15, that is, (1+2+4+8) = 15. In basic terms, the maximum node number  possible for this height h is (20 + 21 + 22+….2h) = 2h+1 -1.

Now, for the minimum node number that is possible at this height h, it comes as equal to h+1.

If there are a minimum number of nodes, then the height of a binary tree would stand aa maximum. On the other hand, when there is a number of a node at its maximum, then the binary tree m height will be minimum. If there exists around ‘n’ number nodes in a binary tree, here is a calculation to clarify the binary tree definition.

The tree’s minimum height is computed as:

n = 2h+1 -1

n+1 = 2h+1

Taking log for both sides now,

log2(n+1) = log2(2h+1)

log2(n+1) = h+1

h = log2(n+1) – 1

The highest height will be computed as:

n = h+1

h= n-1

Binary Tree Components

There are three binary tree components. Every binary tree node has these three components associated with it. It becomes an essential concept for programmers to understand these three binary tree components: 

1. Data element
2. Pointer to left subtree
3. Pointer to right subtree

Source

These three binary tree components represent a node. The data resides in the middle. The left pointer points to the child node, forming the left sub-tree. The right pointer points to the child node at its right, creating the right subtree. 

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Binary Tree Definition: An in-depth analysis

In case there exists n number of nodes in any binary tree, the height stands given by log n logn. This is due to the simple reason that for any given node in any binary tree, there will be two child nodes at the most. This drives us to an explanation to define binary tree: for every level or every height of any binary tree, the node number must be the same as around half the node numbers present at the next level.

In the form of an answer to define binary tree, the node number at every level is close to double the node number at its previous level. We hope this clears the answer to what is binary tree!

It means, that when a binary tree comes with height h, the number of nodes for define binary tree is  n = (2 ^ 0) + (2 ^ 1) + (2 ^ 2) + (2 ^ 3) + ….. + (2 ^ (h-1))(20)+(21)+(22)+(23)+…..+(2(h−1))

From the `mathematical induction point for what is a binary tree, this is what we know-

(2 ^ 0) + (2 ^ 1) + (2 ^ 2) + (2 ^ 3) + ….. + (2 ^ {(h-1)}) = (2 ^ h)-1(20)+(21)+(22)+(23)+…..+(2(h−1))=(2h)−1

Hence,

(2 ^ h)-1 = n => 2 ^ h = n + 1 => h = log2(n+1)(2h)−1=n=>2h=n+1=>h=log2(n+1)

Therefore, the minimum height for a binary tree is roughly equal to log(n) roughly. This helps you better understand what is a binary tree.

Also, the minimum number of nodes that are possible at height h of the binary tree can be known by h+1.

If the binary tree comes with an L number for leaf nodes, the height is represented by L + 1.

Types of Binary Trees

Binary trees come in various types, each suited for specific applications. Full Binary Trees have every non-leaf node with exactly two children. Complete Binary Trees are fully populated at all levels except possibly the last, which is filled from left to right. Perfect Binary Trees have all levels fully filled, with all leaves at the same level. Balanced Binary Trees minimize the height difference between left and right subtrees to optimize search operations. Finally, Binary Search Trees (BST) organize data such that each node’s left subtree contains only lesser values, and the right subtree only greater values, facilitating efficient search and navigation.

There are various types of binary trees, and each of these binary tree types has unique characteristics. Here are each of the binary tree types in detail:

1. Full Binary Tree

It is a special kind of a binary tree that has either zero children or two children. It means that all the nodes in that binary tree should either have two child nodes of its parent node or the parent node is itself the leaf node or the external node. 

In other words, a full binary tree is a unique binary tree where every node except the external node has two children. When it holds a single child, such a binary tree will not be a full binary tree. Here, the quantity of leaf nodes is equal to the number of internal nodes plus one. The equation is like L=I+1, where L is the number of leaf nodes, and I is the number of internal nodes.

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Here is the structure of a full binary tree:

2. Complete Binary Tree

A complete binary tree is another specific type of binary tree where all the tree levels are filled entirely with nodes, except the lowest level of the tree. Also, in the last or the lowest level of this binary tree, every node should possibly reside on the left side. Here is the structure of a complete binary tree:

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3. Perfect Binary Tree

A binary tree is said to be ‘perfect’ if all the internal nodes have strictly two children, and every external or leaf node is at the same level or same depth within a tree. A perfect binary tree having height ‘h’ has 2h – 1 node. Here is the structure of a perfect binary tree:

4. Balanced Binary Tree

A binary tree is said to be ‘balanced’ if the tree height is O(logN), where ‘N’ is the number of nodes. In a balanced binary tree, the height of the left and the right subtrees of each node should vary by at most one. An AVL Tree and a Red-Black Tree are some common examples of data structure that can generate a balanced binary search tree. Here is an example of a balanced binary tree:

5. Degenerate Binary Tree 

A binary tree is said to be a degenerate binary tree or pathological binary tree if every internal node has only a single child. Such trees are similar to a linked list performance-wise. Here is an example of a degenerate binary tree:

Benefits of a Binary Tree

• The search operation in a binary tree is faster as compared to other trees
• Only two traversals are enough to provide the elements in sorted order
• It is easy to pick up the maximum and minimum elements
• Graph traversal also uses binary trees
• Converting different postfix and prefix expressions are possible using binary trees

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Special Types of Binary Trees

Binary trees can also be grouped according to node values. The types of binary tree according to node structure include the following:

• Binary Search Tree

A binary search tree comes with the following properties:

• In the left subtree of any node, you will find nodes with keys smaller than the node’s key.
• The right subtree of any node will include nodes with keys larger than the node’s key.
• The left, as well as the right subtree, will be types of binary search tree. 

• AVL Tree

An AVL binary tree in DSA is self-balanced. In such a tree, the difference between the heights of the left and right subtrees for all nodes cannot be greater than one. So, the nodes in the right as well as left subtrees of the AVL tree will be one or less than that.

• Red Black Tree

If you want us to explain the binary tree and its types, the red-black tree will definitely find a mention. This kind of binary tree is self-balancing, with each node having an extra bit. The extra bit gets represented as either black or red.

The colors in a red black binary tree in data structure are useful for keeping the whole tree balanced during deletions and insertions. The balance of a red black tree won’t be perfect. But these binary trees are perfect for bringing down the search time.

• B- Tree

A B- Tree is a type of self-balanced search tree in data structures. These binary trees support smooth access, deletion, and insertion of data items. B- trees are particularly common in file systems and databases.

Among the different types of binary tree, a B- tree helps with efficient storage and retrieval of large volumes of data. A fixed maximum degree or order is a key characteristic of a B- tree. This fixed value helps determine the total number of child nodes in a parent node.

The nodes present in a B- binary tree can include several keys and child nodes. The keys of a B- binary tree in algorithm design can help in indexing and locating data items. 

• B+ Tree

A binary tree in data structure can also be classified as B+, which is one variant of the B- tree. Since a B+ tree comes with a fixed maximum degree, it enables efficient insertion, access, and deletion of data items. But a B+ binary tree includes all data items inside the leaf nodes. 

The internal nodes of a B+ binary tree only include keys for locating and indexing data items. Due to this design, searches using a B+ tree will be a lot faster, and you will also be able to access data items sequentially. Moreover, the leaf nodes of a binary tree remain together in a linked list. 

• Segment Tree

If you look into a binary tree and its types, you will come across one category called the segment or statistic tree. This type of binary tree is usually responsible for storing information related to different segments or intervals. With a segment tree, you will be able to perform querying of the stored segments in a specific point.

Among the different types of binary tree in data structure, you will realize that a segment tree is static. Therefore, you won’t be able to modify the structure of a segment tree after it has been built.

Applications of Binary Tree in Data Structure

If you want to read more on binary tree in data structure, you should learn about their applications. Binary search trees are quite suitable for the following purposes:

• Search Algorithms: An algorithm for binary search tree can efficiently find a specific element. The search can be executed in O u(log n) time complexity, where n defines the number of nodes. A binary search tree is often useful for quickly finding particular elements in a sorted list. 
• Database Systems: With each node of a binary tree representing a record, data can be stored in a database system. As a result, search operations may be completed quickly, and the database system can manage massive volumes of data.
• Decision Trees: Binary trees are a sort of machine learning technique that may be used to create decision trees. These decision trees are highly useful for regression analysis and classification.
• File Systems: File systems can be implemented using binary trees, in which every node corresponds to a directory or file. This enables quick and easy file system browsing and searching.
• Compression Algorithms: An algorithm for binary search tree in data structure can be useful for Huffman coding. A compression algorithm is responsible for assigning variable-length codes to characters according to their occurrence frequency in the input data. 
• Game AI: Game AI can be implemented using binary trees, where every node indicates a potential move in the game. The optimal move can be found by the AI algorithm searching the tree.
• Sorting Algorithms: An algorithm of binary tree can also be used for efficient sorting. For instance, the search tree sort and heap sort are quite beneficial. 

What are some applications of binary tree?

Binary trees are versatile in computing applications: 1. Hierarchical Data Representation: Ideal for organizing structures like file systems. 2. Database Indexing: Used in databases for quick data retrieval. 3. Priority Queues: Binary heaps implement queues where elements are processed by priority. 4. Expression Parsing: Crucial in compilers for building parse trees that aid in syntax analysis. 5. Network Routing Algorithms: Facilitate efficient data packet routing in network protocols. Each application utilizes the tree structure to efficiently manage and organize data.

Why Should You Use a Binary Tree in Data Structure?

Once you learn about a binary tree and its types, you should try to figure out the benefits of these structures. Some key advantages of using a binary tree model include:

• Ordered Traversal: Binary trees are structured in such a way that you will succeed in traversing them in a particular order, such as post-order, in-order, and pre-order. As a result, you will succeed in performing operations on the nodes in a particular order. For instance, you will be able to easily print nodes in a sorted order. 
• Efficient Searching: A binary tree in data structure can be efficiently used to find a particular element. Each node comes with a maximum of two child nodes. So, search operations can be easily performed with the O(log n) time complexity. 
• Fast Insertion and Deletion: Insertions and deletions can be done with binary trees in O(log n) time complexity. They are also a wise option for applications like database systems that need dynamic data structures.
• Memory Efficient: Since binary trees only need two child pointers per node, they are comparatively memory-efficient when compared to other tree designs. This implies that they can be utilized to maintain effective search functions even when storing substantial volumes of data in memory.
• Valuable for Sorting: If you understand the binary tree terminology in data structure, you will realize that it is extremely efficient for sorting. Therefore, you will find binary trees to be highly beneficial for heap sort and similar operations. 
• Easy to Implement: It is quite simple and easy to understand and implement binary tree structures. That’s why binary tree algorithms are highly suitable for a large number of real-life applications. 

Disadvantages of Binary Tree Structures

While a binary tree in data structure is highly beneficial, it also has some shortcomings. A few reasons why binary trees might not be beneficial include:

• Limited Structure: Every binary tree comes with a maximum of two children in each node. While it is a boon in many ways and makes these structures memory efficient, it is also a disadvantage. Due to their limited structure, binary trees cannot be used in certain cases. For instance, some trees require each node to have more than two children. In that case, a different tree format needs to be used in a data structure. 
• Space Inefficiency: Binary trees are not as space efficient as some other types of data structures. Every node needs two child pointers. So, if it’s a large binary tree, a significant amount of memory will be required. 
• Slow Performances: Binary trees are responsible for extremely slow performances in the worst-case scenarios. The worst-case scenario might even degenerate a binary tree. If that happens, every node will end up with just one child instead of two. As a result, search operations will degrade. 
• Unbalanced Trees: In an unbalanced binary tree, one subtree appears to be considerably larger than the other. This difference can easily render search operations inefficient. The difference is even more prominent when the tree isn’t properly balanced, or data is inserted within it in a non-random manner. 
• Complex Balancing Algorithms: Several balancing algorithms can be used to keep a binary tree balanced. But these algorithms are extremely difficult to implement. Some of these algorithms also demand extra overhead, which makes them incapable of certain applications. 

Operations to Perform on a Binary Tree

Some basic operations that can be implemented on a binary tree include:

• Insertion of an element
• Removal of an element
• Looking for an element
• Deletion of an element
• Traversing an element (You can perform four types of traversals in a binary tree structure.)

A binary tree is also suitable for performing a host of auxiliary operations. Some auxiliary operations to implement on a binary tree include:

• Detecting the height of the tree
• Figuring out the level of the tree
• Determining the right size of the whole tree

Conclusion

The binary tree is one of the most widely used trees in the data structure. Each of the binary tree types has its unique features. These data structures have specific requirements in applied computer science. We hope this article about types of binary trees was helpful. upGrad offers various courses in data science, machine learning, big data, and more.

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