Binary Tree | Binary Tree Properties

Binary Tree-

 

Before you go through this article, make sure that you gone through the previous article on Binary Trees.

 

We have discussed-

  • Binary tree is a special tree data structure.
  • In a binary tree, each node can have at most 2 children.
  • There are following types of binary trees-

 

 

In this article, we will discuss properties of binary trees.

 

Binary Tree Properties-

 

Important properties of binary trees are-

 

Property-01:

 

Minimum number of nodes in a binary tree of height H

= H + 1

 

Example-

 

To construct a binary tree of height = 4, we need at least 4 + 1 = 5 nodes.

 

 

Property-02:

 

Maximum number of nodes in a binary tree of height H

= 2H+1 – 1

 

Example-

 

Maximum number of nodes in a binary tree of height 3

= 23+1 – 1

= 16 – 1

= 15 nodes

Thus, in a binary tree of height = 3, maximum number of nodes that can be inserted = 15.

 

 

We can not insert more number of nodes in this binary tree.

 

Property-03:

 

Total Number of leaf nodes in a Binary Tree

= Total Number of Degree-2 nodes + 1

 

Example-

 

Consider the following binary tree-

 

 

Here,

  • Number of leaf nodes = 3
  • Number of degree-2 nodes = 2

 

Clearly, number of leaf nodes is one greater than number of degree-2 nodes.

This verifies the above relation.

 

NOTE

It is interesting to note that-

Number of leaf nodes in any binary tree depends only on the number of degree-2 nodes.

 

Property-04:

 

Maximum number of nodes at any level ‘L’ in a binary tree

= 2L

 

Example-

 

Maximum number of nodes at level-2 in a binary tree

= 22

= 4

Thus, in a binary tree, maximum number of nodes that can be present at level-2 = 4.

 

 

To gain better understanding about Binary Tree Properties,

Watch this Video Lecture

 

PRACTICE PROBLEMS BASED ON BINARY TREE PROPERTIES-

 

Problem-01:

 

A binary tree T has n leaf nodes. The number of nodes of degree-2 in T is ______?

  1. log2n
  2. n-1
  3. n
  4. 2n

 

Solution-

 

Using property-3, we have-

Number of degree-2 nodes

= Number of leaf nodes – 1

= n – 1

 

Thus, Option (B) is correct.

 

Problem-02:

 

In a binary tree, for every node the difference between the number of nodes in the left and right subtrees is at most 2. If the height of the tree is h > 0, then the minimum number of nodes in the tree is ______?

  1. 2h-1
  2. 2h-1 + 1
  3. 2h – 1
  4. 2h

 

Solution-

 

Let us assume any random value of h. Let h = 3.

Then the given options reduce to-

  1. 4
  2. 5
  3. 7
  4. 8

 

Now, consider the following binary tree with height h = 3-

 

 

  • This binary tree satisfies the question constraints.
  • It is constructed using minimum number of nodes.

 

Thus, Option (B) is correct.

 

Problem-03:

 

In a binary tree, the number of internal nodes of degree-1 is 5 and the number of internal nodes of degree-2 is 10. The number of leaf nodes in the binary tree is ______?

  1. 10
  2. 11
  3. 12
  4. 15

 

Solution-

 

Using property-3, we have-

Number of leaf nodes in a binary tree

= Number of degree-2 nodes + 1

= 10 + 1

= 11

 

Thus, Option (B) is correct.

 

Problem-04:

 

The height of a binary tree is the maximum number of edges in any root to leaf path. The maximum number of nodes in a binary tree of height h is ______?

  1. 2h
  2. 2h-1 – 1
  3. 2h+1 – 1
  4. 2h+1

 

Solution-

 

Using property-2, Option (C) is correct.

 

Problem-05:

 

A binary tree T has 20 leaves. The number of nodes in T having 2 children is ______?

 

Solution-

 

Using property-3, correct answer is 19.

 

To watch video solutions and practice more problems,

Watch this Video Lecture

 

Download Handwritten Notes Here-

 

 

Next Article- Binary Tree Traversal

 

Get more notes and other study material of Data Structures.

Watch video lectures by visiting our YouTube channel LearnVidFun.

Summary
Binary Tree | Binary Tree Properties
Article Name
Binary Tree | Binary Tree Properties
Description
Binary tree is a special tree data structure. There are various types of binary trees. Binary Tree Properties are given. If height of binary tree = H then, minimum number of nodes in binary tree H+1.
Author
Publisher Name
Gate Vidyalay
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