Q12Data Structures

The problem provides the explicit Pre-order traversal of a Binary Search Tree (BST): 20, 5, 3, 4, 10, 15, 30, 25, 40 and requests the corresponding Post-order traversal. Crucially, because this is explicitly stated to be a Binary Search Tree (BST), we possess a hidden, secondary piece of information: the In-order traversal of any valid BST is mathematically guaranteed to be the strictly ascending, sorted order of its constituent nodes.

Step 1: Ascertaining the In-order Sequence

By extracting the nodes from the provided Pre-order list and simply numerically sorting them, we immediately derive the required In-order sequence.

In-order (Sorted): 3, 4, 5, 10, 15, 20, 25, 30, 40

Pre-order (Given): 20, 5, 3, 4, 10, 15, 30, 25, 40

Step 2: Logical Tree Reconstruction

Using the Pre-order sequence to identify roots, and the In-order sequence to define spatial boundaries, we reconstruct the physical tree topology:

• Root is 20. Left subtree elements: {3, 4, 5, 10, 15}. Right subtree elements: {25, 30, 40}.
• Left Subtree (5, 3, 4, 10, 15): Root is 5. Its left child is 3. The right child of 3 is 4. The right child of 5 is 10. The right child of 10 is 15.
• Right Subtree (30, 25, 40): Root is 30. Its left child is 25. Its right child is 40.

Step 3: Generating the Post-order Sequence

With the tree perfectly reconstructed, we can now execute a strict Post-order traversal (Left, Right, Root).

Visiting the left subtree of 20 (rooted at 5): Leftmost is 3, but it has a right child 4. So we visit 4, then 3. Then right subtree of 5 (rooted at 10): Left is empty, right is 15. So we visit 15, then 10. Then the root 5. Sequence so far: 4, 3, 15, 10, 5.

Visiting the right subtree of 20 (rooted at 30): Left child is 25. Right child is 40. Root is 30. Sequence so far: 25, 40, 30.

Finally, visit the absolute root: 20.

The definitively calculated, complete Post-order sequence is: 4, 3, 15, 10, 5, 25, 40, 30, 20.