Using an array-based representation of a binary tree, the following array represents a binary tree.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

59 39 75 48 70 80 44 68 90

What is the postorder traversal of the binary tree that is created in (i).

What is the parent of 68?

Question

Using an array-based representation of a binary tree, the following array represents a binary tree.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

59	39	75		48	70	80			44		68			90

What is the postorder traversal of the binary tree that is created in (i).

What is the parent of 68?

Accepted Answer

```python # Define a function to perform postorder traversal on a binary tree represented as an array def postorder_traversal(tree): # Helper function to traverse the tree def traverse(index): # Base case: if index is out of the range of the tree, return an empty list if index >= len(tree) or tree[index] is None: return [] # Calculate the index of the left and right child in the array left_index = 2 * index + 1 right_index = 2 * index + 2 # Recursively traverse the left subtree, then the right subtree, and append the current node's value return traverse(left_index) + traverse(right_index) + [tree[index]] # Start the traversal from the root of the tree, which is at index 0 return traverse(0) # Define a function to find the parent of a given node in a binary tree represented as an array def find_parent(tree, node): # Try to find the index of the node in the array try: node_index = tree.index(node) except ValueError: # If the node is not in the tree, return None return None # Calculate the index of the parent node parent_index = (node_index - 1) // 2 # If the parent index is out of range, the node is the root and has no parent if parent_index < 0: return None # Return the value of the parent node return tree[parent_index] ``` ```python # Main function that tests postorder_traversal and find_parent functions def main(): # Define the binary tree as an array tree = [59, 39, 75, 48, 70, 80, 44, 68, 90] # Test the postorder_traversal function # Use case 1: Postorder traversal of the given tree assert postorder_traversal(tree) == [48, 39, 44, 68, 70, 90, 80, 75, 59] # Test the find_parent function # Use case 2: Find the parent of node 68 assert find_parent(tree, 68) == 70 # Call the main function to run the tests main() ``` ∻Explanation∻ ⚹The postorder_traversal function recursively visits the left subtree, then the right subtree, and finally the root node for each subtree. The find_parent function calculates the parent's index and returns its value.⚹ ∻Step-by-step Instruction∻ ‖ ▪ ⋮ Define the postorder_traversal function to perform postorder traversal on a binary tree represented as an array. ‖ ‖ ▪ ⋮ Implement a recursive helper function within postorder_traversal to handle the traversal logic. ‖ ‖ ▪ ⋮ Define the find_parent function to find the parent of a given node in the binary tree array. ‖ ‖ ▪ ⋮ Use the index of the node to calculate the index of the parent in the array. ‖ ‖ ▪ ⋮ Define concrete use cases and testing cases for both functions: postorder_traversal should return the correct postorder sequence, and find_parent should return the correct parent node value. ‖ ∻Time Complexity∻ ⚹The time complexity of postorder_traversal is O(n) because it visits each node exactly once. The time complexity of find_parent is O(n) in the worst case due to the use of the index method.⚹ ∻Space Complexity∻ ⚹The space complexity of postorder_traversal is O(n) due to the recursive call stack and the output list. The space complexity of find_parent is O(1) as it only uses a constant amount of extra space.⚹

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