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Binary Tree in Data Structures

What is a Binary Tree in Data Structures?

The term binary itself means bi, which implies two of anything. So very clearly, we know we present the trees in the form of roots and nodes, and so the binary tree is the one that has a minimum of two children. The name provides us an idea where we get to know that the nodes present in the tree can either have 0, 1, or 2 children. In the coming section, we will understand the working of binary trees in more depth by looking at an example.

Given that, it is an example of a binary tree.

By looking at the picture, we know that the above picture is a binary tree as every node present in the tree contains almost two given children. We will now describe and present the above picture in a more precise manner. 

In the above tree, we can observe that node X has two pointers present on its left side as the well right side. The node Y has further divided pointers present each on the left and right side as well, so it is known that it contains the children nodes, but we can observe that the Z, T, and U pointers don’t have any node originating from them. Hence, they are called the leaf nodes and are known to be containing the null pointers on both sides as well. There are some properties for creating and implementing the binary tree, and we have to follow them strictly.

The binary search tree typically divides the trees into two levels that are the right subtree and the left subtree. It is divided as follows: -

Left_tree (key) < node (key) ≤ right_tree

A binary tree generally acquires the following features:-

  • A pointer
  • Data
  • Pointer for the left node or child
  • Pointer for the right node or child

Minimum height of the binary tree

We clearly know that.

n = 2h+1 – 1

n + 1 = 2h+1 

putting log on the both the sides:

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

log2(n+1) = h + 1

h = log2(n + 1) – 1

So, the maximum height can be calculated as:

n = h + 1

h = n – 1

Types of binary trees

In this section of the article, we are going to discuss the various types of binary trees present in the data structures and their properties through which we can implement them.

1.Full Binary Tree

The full binary tree is also very well known as the strict or proper binary tree. If we are given a binary tree, then to figure if whether it is a full binary tree or not, we must keep in mind that the given tree must have at least zero or two children. Otherwise, it is not a full binary tree. It is also described as the tree in which the nodes consist of two children except the leaf nodes. 

Properties of the full binary tree

  • In a full binary tree, the minimal number of nodes is generally calculated by 2*h – 1.
  • In a full binary tree, the height of the structure of a full binary tree is generally calculated by log2 (n + 1) – 1.
  • The total number of nodes present in the full binary tree appears to be the same as in the normal binary tree, which is 2h+1 – 1.
  • There’s a term called the maximum height of the tree and it can be calculated by: -
    • n = 2*h – 1
    • n + 1 = 2*h 
    • h = n + 1 / 2
  • The total number of leaf nodes present in the full binary tree is mainly equal to the total number of initial nodes with just a mere difference that to count the initial nodes. We have to just plus one the total number of leaf nodes.

2.Complete Binary Tree

The complete binary tree is a binary tree that is intact, which implies that all the nodes that are present in the complete binary tree are almost full except for the part in the last segment. In the last segment or the level, all the nodes that are present there should be as left in the direction as possible. In this type of binary tree, the nodes are subjected to be added from the left direction. 

Properties of the complete binary tree

  • In a complete binary tree, the minimal number of nodes is generally calculated by 2h.
  • In a complete binary tree, the height of the structure of a full binary tree is generally calculated by log2 (n + 1) – 1.
  • There’s a term called the maximum height of the tree, and it can be calculated by log2n.
  • There’s a term called the maximum number of nodes in the tree, and it can be calculated by 2h+1 – 1.

3. Perfect Binary tree

A tree, especially a binary tree, is known as a perfect binary tree only and only when the internal nodes or initial nodes present in the tree consist of two children. Also, it is a keynote that all the leaf nodes present should be at the same segment or level. 

Properties of the perfect binary tree

  • The total number of nodes present in the perfect binary tree is 2h + 1 – 1.
  • The total amount of leaf nodes present in the perfect binary tree is calculated by L = 2h.
  • The minimal height of the perfect binary tree is of the value O (log2(n)).

4.Degenerate Binary tree

A tree, especially a binary tree, is known as a degenerate binary tree only when all the internal nodes or initial nodes present in the tree have only one child originating from them. They are called degenerate because they have only one node coming out from them, so they are minimized. They are also sometimes referred to as the right or left-skewed binary tree as they have only one node originating from them and too in a specific direction.

Properties of the degenerate binary tree

  • In a degenerated binary tree, the minimal number of nodes is generally calculated as 2h+1 – 1.
  • It contains only one child at a time originating from a particular node.

5. Balanced Binary tree

A tree, especially a binary tree, is known as a balanced binary tree when it’s a kind of tree where both the left and right sides are differentiated by one. We know that the Red-black tree and AVL tree both are examples and illustrations of the balanced binary tree. It is also termed or described as the balanced or systematic binary tree when the difference between both the sub trees is 0. When the difference between the left and the right subtree is more than one (1), then it is not considered the balanced binary tree.

Properties of the balanced binary tree

  • The first and foremost condition is to keep a check that the left and right sub trees should not have a difference of more than one. 
  • The right subtree is in a systematic order or says balanced.
  • The same goes for the left subtree as well. It is also in a systematic order or says balanced.

Algorithm

struct node {
int number;
struct node *leftchildren;
struct node *rightchildren;
};

Complexity

In this section of the article, we are going to discuss the complexity of the binary tree in the best, worst and average cases. We will also encounter the space complexity of the same in this section. 

Case  Time  
Best caseO(log n)
Average caseO(N0.5)
Worst caseO(N)
  • BEST CASE

In a binary tree, the best-case occurs when the tree can be further divided into two sub-trees that are perfectly balanced. It also appears the best case when the difference between both sides is one. The best-case complexity of the binary tree is big O of N, which is written as O(log n).

  • AVERAGE CASE

The average case usually occurs when we know the height of the perfectly balanced binary tree. If the height of the binary tree is O(N0.5), then the average run time can be easily calculated through this. The average case complexity of the binary tree is big O of N, which is written as O(N0.5).

  • WORST CASE

The worst-case complexity occurs when the given tree is completely and wholly skewed in one specific dimension, which can be either left or right. The worst-case complexity of the binary tree is big O of N, which is written as O(N).

Space Complexity

Space complexity is nothing but the whole amount of memory or space that a specific algorithm or program takes at the time of the execution of that program. The space complexity of the Binary tree is said to be O(1).



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