# Evaluate Postfix Notations

To explain the concept of evaluating postfix notations, let's start with a brief overview:

Postfix notation, also known as Reverse Polish Notation (RPN), is a mathematical notation in which operators are written after their operands. In postfix notation, there is no need for parentheses to indicate the order of operations. Instead, the order of operations is determined by the position of the operators.

To evaluate an expression in postfix notation, we can use a stack to store the operands. We iterate through each token in the expression from left to right. If the token is an operand, we push it onto the stack. If the token is an operator, we pop the required number of operands from the stack, perform the operation, and push the result back onto the stack. After processing all the tokens, the final result will be at the top of the stack.

Let's solve a question from LeetCode using this concept: "Evaluate Reverse Polish Notation" (https://leetcode.com/problems/evaluate-reverse-polish-notation/).

Problem Statement: Given an arithmetic expression in postfix notation, evaluate it and return the result.

Example: Input: tokens = ["2", "1", "+", "3", "*"] Output: 9 Explanation: ((2 + 1) * 3) = 9

Now, let's write the code to solve this problem using the concept of evaluating postfix notations:

```
def evalPostfix(tokens):
operand_stack = []
for token in tokens:
if token.isdigit() or (token[0] == '-' and token[1:].isdigit()):
# If the token is a number, push it onto the operand stack
operand_stack.append(int(token))
else:
# If the token is an operator, pop the required number of operands, perform the operation, and push the result back onto the stack
operand2 = operand_stack.pop()
operand1 = operand_stack.pop()
if token == '+':
result = operand1 + operand2
elif token == '-':
result = operand1 - operand2
elif token == '*':
result = operand1 * operand2
else:
# In Python 3, division of two integers returns a float
result = int(operand1 / operand2)
operand_stack.append(result)
# The operand stack will contain the final result of the expression
return operand_stack[0]
```

The `evalPostfix`

function takes a list of tokens in postfix notation as input and evaluates the expression using the stack-based approach. It iterates through each token and performs the required operations. Finally, it returns the result obtained from the operand stack.

Time Complexity: The time complexity of this algorithm is O(n), where n is the number of tokens in the input expression. We iterate through each token once, and each operation takes constant time.

Space Complexity: The space complexity of this algorithm is O(n), where n is the number of tokens in the input expression. We use a stack to store the operands, and the size of the stack can be at most n/2 in the worst case (when all tokens are operands).

Overall, evaluating postfix notations using a stack-based approach provides an efficient way to compute arithmetic expressions, with a linear time complexity.

The concept of evaluating postfix notations involves calculating the result of an arithmetic expression that is written in postfix notation. Postfix notation, also known as Reverse Polish Notation (RPN), is a mathematical notation where operators are placed after their operands.

To evaluate an expression in postfix notation, we can use a stack to store the operands. We iterate through each element of the expression from left to right. If an element is an operand, we push it onto the stack. If an element is an operator, we pop the required number of operands from the stack, perform the operation, and push the result back onto the stack. After processing all the elements, the final result will be at the top of the stack.

Let's illustrate this concept with an example in Python:

```
def evaluate_postfix(tokens):
stack = []
for token in tokens:
if token.isdigit():
# If the token is a digit, convert it to an integer and push it onto the stack
stack.append(int(token))
else:
# If the token is an operator, pop the required number of operands, perform the operation, and push the result back onto the stack
operand2 = stack.pop()
operand1 = stack.pop()
if token == '+':
result = operand1 + operand2
elif token == '-':
result = operand1 - operand2
elif token == '*':
result = operand1 * operand2
elif token == '/':
# In Python, division of two integers returns a float, so we convert it back to an integer
result = int(operand1 / operand2)
# Push the result back onto the stack
stack.append(result)
# The final result will be at the top of the stack
return stack[0]
```

In this code, the `evaluate_postfix`

function takes a list of tokens as input, representing an arithmetic expression in postfix notation. We initialize an empty stack and iterate through each token. If the token is a digit, we convert it to an integer and push it onto the stack. If the token is an operator, we pop the required number of operands, perform the operation, and push the result back onto the stack. Finally, we return the result from the top of the stack.

The time complexity of this algorithm is O(n), where n is the number of tokens in the expression. We iterate through each token once, and each operation (push, pop, arithmetic) takes constant time. The space complexity is also O(n) because we use a stack to store the operands, and in the worst case, the stack can hold all the operands.

Overall, evaluating postfix notations using a stack-based approach provides an efficient and straightforward method to compute arithmetic expressions.