The image contains several problems related to Trees, Sorting, and Graphs. I'll break each question into parts and provide solutions or explanations. Let me address them step by step:

1. Create a Binary Tree

• In-order traversal: B I D A C G E H F
• Post-order traversal: I D B G C H F E A

Using the given traversals, we can construct the binary tree. I'll start from the post-order traversal (last element = root) and use in-order for placement.

Steps to Construct:

1. Last node in post-order (A) is the root.
2. Find A in in-order traversal. Left subtree = B I D, right subtree = C G E H F.
3. Repeat for left and right subtrees.

Binary Tree:

        A
      /   \
     B     C
      \      \
       D      G
      /      / \
     I      E   H
                /
               F

2. Write an Algorithm for Merge Sort and Apply

Array: 45, 32, 65, 76, 23, 1, 2, 54, 67, 22, 07

Merge Sort Algorithm:

1. Divide: Split the array into halves recursively until each sub-array has one element.
2. Merge: Combine sorted sub-arrays to form a single sorted array.

Sorted Array:
Applying the algorithm gives: 1, 2, 7, 22, 23, 32, 45, 54, 65, 67, 76

3. Write Short Notes

1. Complete Binary Tree:

   • A binary tree in which all levels are completely filled, except possibly the last level, which is filled from left to right.

2. Extended Binary Tree:

   • Also called a 2-tree, an extended binary tree adds external nodes (null pointers) to a binary tree, making each node have exactly two children (either real or external).

4. Find the Minimum Spanning Tree (MST) Using Prim's Algorithm

Graph Details:
Vertices = {A, B, C, D, E}, Edges with weights as shown.

Steps (Prim’s Algorithm):

1. Start with vertex A.
2. Add the smallest edge connecting A to another vertex (A - B, weight = 1).
3. Continue adding the smallest edge that connects to the growing MST, avoiding cycles.

MST Edges:
A-B (1), B-C (2), C-D (1), D-E (2)
Total Weight: 6.

5. Construct an AVL Tree

Elements: 71, 41, 91, 56, 60, 30, 40, 80, 50, 55

Steps:

1. Start with an empty tree, insert elements one by one, and perform rotations to maintain AVL properties:
   • Balance factor of each node = height(left) - height(right), must be in {-1, 0, 1}.
2. Apply LL, RR, LR, or RL rotations as necessary.

AVL Tree (Final Structure):

           56
         /    \
       41      71
      /  \       \
    30    50      91
        \   \     /
        40   55  80
                /
               60

Would you like detailed steps for any problem, or explanations of how calculations or constructs were derived? Let me know!

Expanding Knowledge

1. How does post-order traversal differ from in-order in recursive implementation?
2. What are the key applications of merge sort in computer science?
3. How do Prim’s Algorithm and Kruskal’s Algorithm compare in efficiency?
4. What is the importance of balance factor in AVL trees?
5. Can we use AVL trees for sorting large datasets? Why or why not?

Tip:

While constructing trees (binary, AVL, or others), always label the steps clearly to visualize rotations or insertions effectively.