请详细回答以下问题：解释什么是递归，并举例说明。请展开论述，分点说明，条理清晰。

Method: POST

URL: https://tokease.cn/v1/chat/completions

Headers:
{
  "Authorization": "Bearer sk-oRd5RE9JhT1bDLQ3lOZgdkESeCuowucrcNX16gZEgHecnxmu",
  "Content-Type": "application/json"
}

Body:
{
  "max_tokens": 300,
  "messages": [
    {
      "role": "user",
      "content": "请详细回答以下问题：解释什么是递归，并举例说明。请展开论述，分点说明，条理清晰。"
    }
  ],
  "model": "deepseek-v4-flash/yd"
}

Headers:
{
  "Content-Length": "1947",
  "Content-Type": "application/json",
  "Date": "Fri, 21 Aug 2026 08:23:13 GMT",
  "Req-Arrive-Time": "1787300592667",
  "Req-Cost-Time": "505",
  "Resp-Start-Time": "1787300593172",
  "Server": "nginx",
  "Set-Cookie": "route=7041e75fbdb03d21cbe6a8d8f36f7b94; Path=/",
  "Vary": "Accept-Encoding",
  "Via": "1.1 Caddy",
  "X-Envoy-Upstream-Service-Time": "495",
  "X-New-Api-Version": "v0.0.0",
  "X-Oneapi-Request-Id": "202608210823126378923458268d9d6SeBhbIvU",
  "X-Request-Id": "44310331-57f1-49fa-a010-6a566011e75a"
}

Body:
{
  "id": "chatcmpl-119d5d33-f851-42c1-b23c-ae90dd08db82",
  "object": "chat.completion",
  "created": 1787300576,
  "model": "deepseek-v4-flash",
  "choices": [
    {
      "index": 0,
      "message": {
        "role": "assistant",
        "content": "### 递归的详细解释\n\n#### 一、什么是递归\n递归是一种编程技术，**函数直接或间接地调用自身**，通过将一个大问题分解为结构相同但规模更小的子问题，逐步简化直到达到一个**基础情况（终止条件）**，从而解决问题。\n\n核心思想是 **“分而治之”** 和 **“自我引用”**。递归包含两个关键部分：\n1. **递归步骤**（Recursive Step）：函数调用自身，每次调用时问题规模缩小。\n2. **基础情况**（Base Case）：不再调用自身，直接返回结果的终止条件，防止无限循环。\n\n---\n\n#### 二、递归的组成部分（详细展开）\n\n| 组成部分 | 作用 | 示例（计算阶乘） |\n|---------|------|----------------|\n| **基础情况** | 最小子问题的直接解，结束递归 | `factorial(0) = 1` |\n| **递归步骤** | 将当前问题转化为更小规模的同类问题 | `factorial(n) = n * factorial(n-1)` |\n| **递归调用** | 函数内部调用自身 | `return n * factorial(n-1)` |\n| **返回值传递** | 将子问题的结果组合成原问题的解 | 每一步的乘法运算 |\n\n---\n\n#### 三、经典举例：计算阶乘（n!）\n\n**问题定义**：n! = n × (n-1)",
        "refusal": null,
        "annotations": null,
        "audio": null,
        "function_call": null,
        "reasoning": null
      },
      "logprobs": null,
      "finish_reason": "length",
      "stop_reason": null,
      "token_ids": null,
      "routed_experts": null
    }
  ],
  "service_tier": null,
  "system_fingerprint": "vllm-0.25.1-dp8-ep-1ccbc6be",
  "usage": {
    "prompt_tokens": 30,
    "total_tokens": 330,
    "completion_tokens": 300,
    "prompt_tokens_details": {
      "cached_tokens": 0,
      "multimodal_tokens": null
    }
  },
  "prompt_logprobs": null,
  "prompt_token_ids": null,
  "prompt_text": null,
  "kv_transfer_params": null,
  "metrics": null
}

{
  "prompt_tokens": 30,
  "total_tokens": 330,
  "completion_tokens": 300,
  "prompt_tokens_details": {
    "cached_tokens": 0,
    "multimodal_tokens": null
  }
}

Method: POST

URL: /v1/chat/completions

Headers:
{
  "Accept": "*/*",
  "Accept-Encoding": "gzip",
  "Authorization": "Bearer sk_r4gAXloJQt74Cp4prb19OItKzoG77lLhjmWT4ei20Ua",
  "Content-Length": "211",
  "Content-Type": "application/json",
  "User-Agent": "Go-http-client/2.0",
  "Via": "2.0 Caddy",
  "X-Forwarded-For": "13.212.110.206",
  "X-Forwarded-Host": "api.tokendiet.cn",
  "X-Forwarded-Proto": "https"
}

Body:
{
  "model": "deepseek-v4-flash-g1",
  "messages": [
    {
      "role": "user",
      "content": "请详细回答以下问题：解释什么是递归，并举例说明。请展开论述，分点说明，条理清晰。"
    }
  ],
  "max_tokens": 300
}

Body:
{
  "choices": [
    {
      "finish_reason": "length",
      "index": 0,
      "logprobs": null,
      "message": {
        "annotations": null,
        "audio": null,
        "content": "### 递归的详细解释\n\n#### 一、什么是递归\n递归是一种编程技术，**函数直接或间接地调用自身**，通过将一个大问题分解为结构相同但规模更小的子问题，逐步简化直到达到一个**基础情况（终止条件）**，从而解决问题。\n\n核心思想是 **“分而治之”** 和 **“自我引用”**。递归包含两个关键部分：\n1. **递归步骤**（Recursive Step）：函数调用自身，每次调用时问题规模缩小。\n2. **基础情况**（Base Case）：不再调用自身，直接返回结果的终止条件，防止无限循环。\n\n---\n\n#### 二、递归的组成部分（详细展开）\n\n| 组成部分 | 作用 | 示例（计算阶乘） |\n|---------|------|----------------|\n| **基础情况** | 最小子问题的直接解，结束递归 | `factorial(0) = 1` |\n| **递归步骤** | 将当前问题转化为更小规模的同类问题 | `factorial(n) = n * factorial(n-1)` |\n| **递归调用** | 函数内部调用自身 | `return n * factorial(n-1)` |\n| **返回值传递** | 将子问题的结果组合成原问题的解 | 每一步的乘法运算 |\n\n---\n\n#### 三、经典举例：计算阶乘（n!）\n\n**问题定义**：n! = n × (n-1)",
        "function_call": null,
        "reasoning": null,
        "refusal": null,
        "role": "assistant"
      },
      "routed_experts": null,
      "stop_reason": null,
      "token_ids": null
    }
  ],
  "created": 1787300576,
  "id": "chatcmpl-gsTrVJ0eVe8einST",
  "kv_transfer_params": null,
  "metrics": null,
  "model": "deepseek-v4-flash-g1",
  "object": "chat.completion",
  "prompt_logprobs": null,
  "prompt_text": null,
  "prompt_token_ids": null,
  "service_tier": null,
  "system_fingerprint": "vllm-0.25.1-dp8-ep-1ccbc6be",
  "usage": {
    "completion_tokens": 300,
    "prompt_tokens": 30,
    "prompt_tokens_details": {
      "cached_tokens": 0,
      "multimodal_tokens": null
    },
    "total_tokens": 330
  }
}