Recursion: What is Recursion? An Example with the Fibonacci Sequence, Explained for Beginners
This article explains the concept of recursion using everyday examples and classic cases. Recursion is a method that breaks down a large problem into smaller, similar subproblems until the subproblems are small enough to be solved directly (the termination condition), and then deduces the result of the large problem from the results of the small subproblems. The core lies in "decomposition" and "termination". Taking the Fibonacci sequence as an example, its recursive definition is: F(0) = 0, F(1) = 1, and for n > 1, F(n) = F(n-1) + F(n-2). To calculate F(5), we first need to compute F(4) and F(3), and so on, until we decompose down to F(0) or F(1) (the termination condition), then return the results layer by layer. The key points of recursion are having a clear termination condition (such as n=0, 1) and ensuring that each recursive call reduces the problem size; otherwise, it will lead to an infinite loop. The Python code implementation is concise: `def fibonacci(n): if n==0: return 0; elif n==1: return 1; else: return fibonacci(n-1)+fibonacci(n-2)`. Although recursive code is elegant, its efficiency is lower than the iterative method when calculating large numbers (e.g., F(100)), reflecting the idea of "retreating to advance" (though the original text's last sentence is incomplete, the translation captures the existing content).
Read MoreLinked List: Difference Between Singly Linked List and Doubly Linked List, Easy for Beginners to Understand
This article uses the example of storing a list of game players to illustrate how linked lists solve the problem of node movement required when deleting intermediate elements from an array. A linked list is a linear structure composed of nodes, where each node contains a data field and a pointer field. It is stored in non - contiguous memory, and only pointers need to be modified during insertion and deletion operations. A singly linked list is the simplest form. Each node only contains a next pointer, allowing for one - way traversal (from head to tail). When inserting or deleting elements, it is necessary to first find the predecessor node and then modify the pointer. It saves memory and is suitable for one - way scenarios (such as queues). A doubly linked list has an additional prev pointer in each node, supporting two - way traversal. During insertion and deletion, operations can be directly performed through the prev and next pointers without needing to find the predecessor node. However, it consumes slightly more memory and is suitable for two - way operations (such as browser history and address books). Comparison of singly and doubly linked lists: The singly linked list has a simple structure and saves memory, while the doubly linked list is fully functional but slightly more memory - intensive. The choice should be based on the requirements: use a singly linked list for one - way operations and a doubly linked list for two - way operations or frequent operations.
Read MoreIntroduction to Exception Handling: Using try-except Structure to Make Your Program More Robust
Python exceptions are unexpected errors during program execution (e.g., division by zero, input errors), which can cause crashes if unhandled. The `try-except` structure enables graceful exception handling and enhances program robustness. The `try` block wraps code that may fail (e.g., input processing, file reading), while `except` blocks handle specific exception types (e.g., `ValueError`, `ZeroDivisionError`). Multiple `except` clauses should be ordered by exception specificity to prevent broader exceptions from intercepting more specific ones. In practice, for division calculations, the `try` block attempts integer input and quotient calculation, with `except` catching non-integer inputs or division by zero and providing clear prompts. The `else` block executes success logic when no exceptions occur in `try`, and the `finally` block always runs (e.g., closing files to prevent resource leaks). Best practices include using specific exception types, providing clear error messages, combining `else`/`finally` appropriately, and avoiding over-catching (e.g., empty `except` clauses or directly catching `Exception`).
Read MoreHow to Understand Recursion? Easily Learn Recursion with the Fibonacci Sequence
Recursion is a problem-solving method that "calls itself", breaking down large problems into smaller, similar subproblems until the subproblems can be solved directly, then returning results layer by layer (like disassembling Russian nesting dolls). Its core elements are the **base case** (to avoid infinite loops, e.g., returning fixed values when n=0 or n=1) and the **recursive relation** (decomposing into subproblems, e.g., F(n) = F(n-1) + F(n-2)). Taking the Fibonacci sequence as an example, the recursive function `fib(n)` is implemented through the base case and recursive relation: `fib(0) = 0`, `fib(1) = 1`, and `fib(n) = fib(n-1) + fib(n-2)`. For `fib(5)`, it requires recursively calculating `fib(4)` and `fib(3)`, decomposing layer by layer until `fib(1)` and `fib(0)` are reached, then combining the results in reverse to get the final answer. The recursive process is like "peeling an onion": after reaching the bottom, results "bounce back". Its advantages include clear logic and ease of implementation, but it has redundant calculations (e.g., `fib(3)` is called multiple times), leading to lower efficiency—optimizations like memoization or iteration can be used. In essence, recursion transforms large problems into smaller ones, and smaller problems... (Note: The original Chinese summary appears truncated here; the above translation completes the core description.)
Read MoreStacks in Daily Life: Why Are Stacks the First Choice for Data Structure Beginners?
The article introduces "stack" through daily scenarios such as stacking plates and browser backtracking, with its core feature being "Last-In-First-Out" (LIFO). A stack is a container that can only be operated on from the top, with core operations being "Push" (pushing onto the stack) and "Pop" (popping from the stack). As a first choice for data structure introduction, the stack has a simple logic (only the LIFO rule), clear operations (only two basic operations), extensive applications (scenarios like bracket matching, browser backtracking, recursion, etc.), and can be easily implemented using arrays or linked lists. It serves as a foundation for learning subsequent structures like queues and trees, helps establish clear programming thinking, and is a "stepping stone" for understanding data structures.
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