Solution Explanation

This problem involves designing an expression tree from a postfix expression and then evaluating it. The solution uses a stack-based approach to build the tree and a recursive approach to evaluate it.

Approach

1. Tree Node Structure: We define a Node class (or similar depending on the language) to represent nodes in the expression tree. Each node stores a value (an operand or operator) and pointers to its left and right children.

2. Building the Tree: The buildTree function iterates through the postfix expression.

   • If an element is an operand (number), a new leaf node is created with that operand as its value.
   • If an element is an operator, two nodes are popped from the stack (representing the operands). A new internal node is created with the operator as its value, and the popped nodes as its left and right children. The new node is then pushed onto the stack.

3. Evaluating the Tree: The evaluate function recursively evaluates the tree.

   • If a node is a leaf node (operand), its value is returned.
   • If a node is an internal node (operator), its left and right children are recursively evaluated. The operator is then applied to the results, and the final result is returned.

Time and Space Complexity

Time Complexity:

• buildTree: O(n), where n is the length of the postfix expression. Each element is processed once.
• evaluate: O(n) in the worst case (a skewed tree). The recursive calls visit each node once.

Space Complexity:

• buildTree: O(n) in the worst case. The stack can grow up to the size of the expression (if the expression is heavily nested).
• evaluate: O(h), where h is the height of the tree. This is due to the recursive call stack. In the worst case (a skewed tree), h can be n. In a balanced tree, h would be log n.

Code Explanation (Python)

The Python code utilizes a MyNode class implementing the abstract Node class. The buildTree method uses a stack to construct the expression tree in a post-order traversal fashion. The evaluate method recursively evaluates the tree, performing the arithmetic operations according to the operator at each internal node.

The code is well-structured with clear comments and follows best practices for readability and maintainability.

Code Explanation (Java)

The Java code mirrors the Python implementation using abstract classes and interfaces. The MyNode class handles the node representation and evaluation, while TreeBuilder constructs the tree using a stack. The isNumeric() helper function efficiently checks if a string is a number.

Code Explanation (C++)

The C++ code also adopts a similar structure to the Python and Java versions, using virtual functions for polymorphism. The use of smart pointers (if needed) would improve memory management. Error handling could be added to the code to handle invalid input scenarios more robustly.