{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "starting-portfolio",
   "metadata": {},
   "source": [
    "# Recursion"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "further-brighton",
   "metadata": {},
   "source": [
    "A recursive algorithm is an algorithm that calls itself. You need a base case so you don't get stuck in an infinite loop."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "sunrise-cyprus",
   "metadata": {},
   "source": [
    "Example: suppose we want to calculate the quantity $n! = n(n-1)(n-2)\\cdots 3 \\cdot 2 \\cdot 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "18459a42",
   "metadata": {},
   "source": [
    "5! = 5 * 4 * 3 * 2 * 1 = 120"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "gorgeous-girlfriend",
   "metadata": {},
   "source": [
    "We'll use the fact that $n! = n \\cdot (n-1)!$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "metallic-psychiatry",
   "metadata": {},
   "outputs": [],
   "source": [
    "# What's wrong with this function?\n",
    "def factorial(n):\n",
    "    return n * factorial(n-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "appropriate-moore",
   "metadata": {},
   "outputs": [
    {
     "ename": "RecursionError",
     "evalue": "maximum recursion depth exceeded",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mRecursionError\u001b[0m                            Traceback (most recent call last)",
      "Cell \u001b[0;32mIn[2], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m \u001b[43mfactorial\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m3\u001b[39;49m\u001b[43m)\u001b[49m\n",
      "Cell \u001b[0;32mIn[1], line 3\u001b[0m, in \u001b[0;36mfactorial\u001b[0;34m(n)\u001b[0m\n\u001b[1;32m      2\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mfactorial\u001b[39m(n):\n\u001b[0;32m----> 3\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m n \u001b[38;5;241m*\u001b[39m \u001b[43mfactorial\u001b[49m\u001b[43m(\u001b[49m\u001b[43mn\u001b[49m\u001b[38;5;241;43m-\u001b[39;49m\u001b[38;5;241;43m1\u001b[39;49m\u001b[43m)\u001b[49m\n",
      "Cell \u001b[0;32mIn[1], line 3\u001b[0m, in \u001b[0;36mfactorial\u001b[0;34m(n)\u001b[0m\n\u001b[1;32m      2\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mfactorial\u001b[39m(n):\n\u001b[0;32m----> 3\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m n \u001b[38;5;241m*\u001b[39m \u001b[43mfactorial\u001b[49m\u001b[43m(\u001b[49m\u001b[43mn\u001b[49m\u001b[38;5;241;43m-\u001b[39;49m\u001b[38;5;241;43m1\u001b[39;49m\u001b[43m)\u001b[49m\n",
      "    \u001b[0;31m[... skipping similar frames: factorial at line 3 (2970 times)]\u001b[0m\n",
      "Cell \u001b[0;32mIn[1], line 3\u001b[0m, in \u001b[0;36mfactorial\u001b[0;34m(n)\u001b[0m\n\u001b[1;32m      2\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mfactorial\u001b[39m(n):\n\u001b[0;32m----> 3\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m n \u001b[38;5;241m*\u001b[39m \u001b[43mfactorial\u001b[49m\u001b[43m(\u001b[49m\u001b[43mn\u001b[49m\u001b[38;5;241;43m-\u001b[39;49m\u001b[38;5;241;43m1\u001b[39;49m\u001b[43m)\u001b[49m\n",
      "\u001b[0;31mRecursionError\u001b[0m: maximum recursion depth exceeded"
     ]
    }
   ],
   "source": [
    "factorial(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "31f6174b",
   "metadata": {},
   "source": [
    "What calls are happening?\n",
    "```py\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "dbbe172d",
   "metadata": {},
   "outputs": [],
   "source": [
    "# What's wrong with this function?\n",
    "def factorial(n):\n",
    "    return n * factorial(n-1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5ba3a205",
   "metadata": {},
   "source": [
    "```py\n",
    "factorial(3)\n",
    " -> 3 * factorial(2)\n",
    " -> 3 * 2 * factorial(1)\n",
    " -> 3 * 2 * 1 * factorial(0)\n",
    " -> 3 * 2 * 1 * 0 * factorial(-1)\n",
    " -> infinite recursion\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "failing-lounge",
   "metadata": {},
   "outputs": [],
   "source": [
    "# We need a base case!\n",
    "def factorial(n):\n",
    "    assert n >= 0 and isinstance(n, int), \"invalid input into factorial\"\n",
    "    if n == 1:\n",
    "        # factorial(1) = 1\n",
    "        return 1\n",
    "    return n * factorial(n-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "acute-delicious",
   "metadata": {},
   "outputs": [
    {
     "ename": "AssertionError",
     "evalue": "invalid input into factorial",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mAssertionError\u001b[0m                            Traceback (most recent call last)",
      "Cell \u001b[0;32mIn[15], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m \u001b[43mfactorial\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m3.5\u001b[39;49m\u001b[43m)\u001b[49m\n",
      "Cell \u001b[0;32mIn[14], line 3\u001b[0m, in \u001b[0;36mfactorial\u001b[0;34m(n)\u001b[0m\n\u001b[1;32m      2\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mfactorial\u001b[39m(n):\n\u001b[0;32m----> 3\u001b[0m     \u001b[38;5;28;01massert\u001b[39;00m n \u001b[38;5;241m>\u001b[39m\u001b[38;5;241m=\u001b[39m \u001b[38;5;241m0\u001b[39m \u001b[38;5;129;01mand\u001b[39;00m \u001b[38;5;28misinstance\u001b[39m(n, \u001b[38;5;28mint\u001b[39m), \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124minvalid input into factorial\u001b[39m\u001b[38;5;124m\"\u001b[39m\n\u001b[1;32m      4\u001b[0m     \u001b[38;5;28;01mif\u001b[39;00m n \u001b[38;5;241m==\u001b[39m \u001b[38;5;241m1\u001b[39m:\n\u001b[1;32m      5\u001b[0m         \u001b[38;5;66;03m# factorial(1) = 1\u001b[39;00m\n\u001b[1;32m      6\u001b[0m         \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;241m1\u001b[39m\n",
      "\u001b[0;31mAssertionError\u001b[0m: invalid input into factorial"
     ]
    }
   ],
   "source": [
    "factorial(3.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "municipal-phoenix",
   "metadata": {},
   "source": [
    "```py\n",
    "factorial(5)\n",
    "5 * factorial(4)\n",
    "5 * (4 * factorial(3))\n",
    "5 * (4 * (3 * factorial(2)))\n",
    "5 * (4 * (3 * (2 * factorial(1))))\n",
    "5 * (4 * (3 * (2 * 1)))\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "functional-pointer",
   "metadata": {},
   "outputs": [
    {
     "ename": "RecursionError",
     "evalue": "maximum recursion depth exceeded",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mRecursionError\u001b[0m                            Traceback (most recent call last)",
      "Cell \u001b[0;32mIn[9], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m \u001b[43mfactorial\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m-\u001b[39;49m\u001b[38;5;241;43m1\u001b[39;49m\u001b[43m)\u001b[49m\n",
      "Cell \u001b[0;32mIn[3], line 6\u001b[0m, in \u001b[0;36mfactorial\u001b[0;34m(n)\u001b[0m\n\u001b[1;32m      3\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m n \u001b[38;5;241m==\u001b[39m \u001b[38;5;241m1\u001b[39m:\n\u001b[1;32m      4\u001b[0m     \u001b[38;5;66;03m# factorial(1) = 1\u001b[39;00m\n\u001b[1;32m      5\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;241m1\u001b[39m\n\u001b[0;32m----> 6\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m n \u001b[38;5;241m*\u001b[39m \u001b[43mfactorial\u001b[49m\u001b[43m(\u001b[49m\u001b[43mn\u001b[49m\u001b[38;5;241;43m-\u001b[39;49m\u001b[38;5;241;43m1\u001b[39;49m\u001b[43m)\u001b[49m\n",
      "Cell \u001b[0;32mIn[3], line 6\u001b[0m, in \u001b[0;36mfactorial\u001b[0;34m(n)\u001b[0m\n\u001b[1;32m      3\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m n \u001b[38;5;241m==\u001b[39m \u001b[38;5;241m1\u001b[39m:\n\u001b[1;32m      4\u001b[0m     \u001b[38;5;66;03m# factorial(1) = 1\u001b[39;00m\n\u001b[1;32m      5\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;241m1\u001b[39m\n\u001b[0;32m----> 6\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m n \u001b[38;5;241m*\u001b[39m \u001b[43mfactorial\u001b[49m\u001b[43m(\u001b[49m\u001b[43mn\u001b[49m\u001b[38;5;241;43m-\u001b[39;49m\u001b[38;5;241;43m1\u001b[39;49m\u001b[43m)\u001b[49m\n",
      "    \u001b[0;31m[... skipping similar frames: factorial at line 6 (2971 times)]\u001b[0m\n",
      "Cell \u001b[0;32mIn[3], line 6\u001b[0m, in \u001b[0;36mfactorial\u001b[0;34m(n)\u001b[0m\n\u001b[1;32m      3\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m n \u001b[38;5;241m==\u001b[39m \u001b[38;5;241m1\u001b[39m:\n\u001b[1;32m      4\u001b[0m     \u001b[38;5;66;03m# factorial(1) = 1\u001b[39;00m\n\u001b[1;32m      5\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;241m1\u001b[39m\n\u001b[0;32m----> 6\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m n \u001b[38;5;241m*\u001b[39m \u001b[43mfactorial\u001b[49m\u001b[43m(\u001b[49m\u001b[43mn\u001b[49m\u001b[38;5;241;43m-\u001b[39;49m\u001b[38;5;241;43m1\u001b[39;49m\u001b[43m)\u001b[49m\n",
      "\u001b[0;31mRecursionError\u001b[0m: maximum recursion depth exceeded"
     ]
    }
   ],
   "source": [
    "factorial(-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "chinese-applicant",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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