## Trees

Question 1 |

The postorder traversal of a binary tree is 8, 9, 6, 7, 4, 5, 2, 3, 1. The inorder traversal of the same tree is 8, 6, 9, 4, 7, 2, 5, 1, 3. The height of a tree is the length of the longest path from the root to any leaf. The height of the binary tree above is _______.

1 | |

2 | |

3 | |

4 |

In – 8 6 9 4 7 2 5 1 3

Post: 8 9 6 7 4 5 2 3

__1__→(root)

In: 8 6 9 4 7 2

__5__→(left subtree)

__13__→(right subtree)

Question 2 |

Let *T *be a tree with 10 vertices. The sum of the degrees of all the vertices in *T* is _________.

18 | |

19 | |

20 | |

21 |

A tree, with 10 vertices, consists n - 1, i.e. 10 - 1 = 9 edges.

Sum of degrees of all vertices = 2(#edges)

= 2(9)

= 18

Question 3 |

The number of ways in which the numbers 1, 2, 3, 4, 5, 6, 7 can be inserted in an empty binary search tree, such that the resulting tree has height 6, is _________.

*Note: The height of a tree with a single node is 0.*

64 | |

65 | |

66 | |

67 |

So to get such tree at each level we should have either maximum or minimum element from remaining numbers till that level. So no. of binary search tree possible is,

1

^{st}level - 2 (maximum or minimum)

⇓

2

^{nd}level - 2

⇓

3

^{rd}level - 2

⇓

4

^{th}level - 2

⇓

5

^{th}level - 2

⇓

6

^{th}level - 2

⇓

7

^{th}level - 2

= 2 × 2 × 2 × 2 × 2 × 2 × 1

= 2

^{6}

= 64

Question 4 |

The height of a tree is the length of the longest root-to-leaf path in it. The maximum and minimum number of nodes in a binary tree of height 5 are

63 and 6, respectively | |

64 and 5, respectively | |

32 and 6, respectively | |

31 and 5, respectively |

2

^{h+1}- 1 = 2

^{5+1}- 1 = 63

Minimum number of nodes in a binary tree of height h is

h + 1 = 5 + 1 = 6

Question 5 |

Consider the following 2-3-4 tree (i.e., B-tree with a minimum degree of two) in which each data item is a letter. The usual alphabetical ordering of letters is used in constructing the tree.

What is the result of inserting G in the above tree?

None of the above |

Insert G at root level:

Question 6 |

A 2-3 tree is tree such that

- (a) all internal nodes have either 2 or 3 children

(b) all paths from root to the leaves have the same length

The number of internal nodes of a 2-3 tree having 9 leaves could be

4 | |

5 | |

6 | |

7 | |

Both A and D |

Where L is leaf node.

So, no. of internal node is 4.

Case 2:

Where L is leaf node.

So, no. of internal node is 7.

Question 7 |

4 | |

8 | |

16 | |

32 |