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Recursion in programming is a technique where a function calls itself in order to solve a problem. It is a powerful concept that allows you to break down complex problems into smaller, more manageable subproblems. Here's a detailed explanation:6 T8 `% j% c- [1 t Y
Key Idea of Recursion
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) f2 J# u( M! Q: X `& d: [A recursive function solves a problem by:0 W) n8 X w$ A3 G7 e
: L' q1 c$ ~ D) m' g2 `$ } Breaking the problem into smaller instances of the same problem.
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$ P7 L! t! N8 k8 L& w5 m) Q2 u/ \ Solving the smallest instance directly (base case).
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Combining the results of smaller instances to solve the larger problem.6 V- C: z3 t: @7 S1 S
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Components of a Recursive Function- I+ _5 W; u9 F6 p7 b; n4 d e0 O2 s
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Base Case:
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0 c! L1 T8 I7 Z1 g3 @6 Y" H This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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It acts as the stopping condition to prevent infinite recursion.9 z) ?6 w4 X' }0 h* [
* u3 R" v1 s. o6 s4 v' s. N Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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- K/ \, r: ]. }1 ~7 b8 `2 i8 a8 X& S2 Q, q Recursive Case:+ e F, a9 [ g, h- E& D
+ @. y* R q0 ~) j T This is where the function calls itself with a smaller or simpler version of the problem.
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Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).$ b. t3 r# [' V4 R# x
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Example: Factorial Calculation9 [& @1 f; ~0 |0 Q" p
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The factorial of a number n (denoted as n!) is the product of all positive integers less than or equal to n. It can be defined recursively as:
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$ @; R6 k% H, {3 l* I% s Base case: 0! = 1$ q3 j. F( H. S
- d# P9 K/ V! z( ~" c Recursive case: n! = n * (n-1)!
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: S# G6 {$ S( d% N- G1 _6 wHere’s how it looks in code (Python):: i( I! q, m1 v
python% B' W, c4 V8 c! r$ Z$ f5 |4 U
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6 v* W& N7 d0 L( F( Y0 T' hdef factorial(n):& V( e- a5 q# B7 G, h
# Base case
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return 1
- I7 j8 d5 q" W4 L$ v/ W* a- R # Recursive case4 o0 f. H, ]3 f
else:# }. ^7 W9 F6 Y" a. m, X$ _: L( E
return n * factorial(n - 1)) T' |/ r3 [) A2 f" k" `6 v
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# Example usage
. S) A3 G S/ m3 uprint(factorial(5)) # Output: 120( P; }$ b, _5 a3 [0 u, b, C
8 u ~2 R: ]4 M- qHow Recursion Works+ O3 {9 D, N' H
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The function keeps calling itself with smaller inputs until it reaches the base case.
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Once the base case is reached, the function starts returning values back up the call stack.( b9 L( y% t5 M( t9 `
$ ?' i3 q$ h7 s+ `; c/ ?- [ These returned values are combined to produce the final result.) j: X3 B, B4 }& R
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For factorial(5):+ ]% m, M" z, g; W- i9 H! A- f( f, S5 u
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0 o$ @% I3 S3 V. j% Ffactorial(5) = 5 * factorial(4) ]" Q" M7 S! h' H; K4 i2 I" V$ K; P
factorial(4) = 4 * factorial(3)
% d+ ^0 ], u. b; X, rfactorial(3) = 3 * factorial(2). h" e+ d6 M- c8 o$ K& k
factorial(2) = 2 * factorial(1), J2 n; m8 @7 i, G9 k8 f
factorial(1) = 1 * factorial(0)
/ D4 j k4 M! }$ qfactorial(0) = 1 # Base case
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Then, the results are combined:2 W4 u9 e: ]8 E. O$ L) O
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" p1 A2 k+ o- B% E- ffactorial(1) = 1 * 1 = 1
; |; D n/ d" f0 H1 O2 r0 Wfactorial(2) = 2 * 1 = 2
+ M+ F ^5 W+ O5 Wfactorial(3) = 3 * 2 = 6
/ l* m1 E' Q$ O/ X9 nfactorial(4) = 4 * 6 = 24
4 Q" l2 H( @" w; cfactorial(5) = 5 * 24 = 120; U1 D! q0 E: O: [( a& s
& f. q0 l- H* U9 H' s2 a) N$ C! IAdvantages of Recursion
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Simplicity: Recursive solutions are often more intuitive and easier to write for problems that have a natural recursive structure (e.g., tree traversals, divide-and-conquer algorithms).
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Readability: Recursive code can be more readable and concise compared to iterative solutions.
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Disadvantages of Recursion
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- L4 ?) f/ u- N9 x" w4 U4 [6 g Performance Overhead: Each recursive call adds a new layer to the call stack, which can lead to high memory usage and potential stack overflow for deep recursion.. f( ^" b( O& y! T8 `1 Z1 l/ d( F
( L" Q9 i' v# X. {1 m Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).) I' Y/ k% B; f8 |" d6 k0 e* w! }
B( r# ]# B# Q6 X% O8 ]When to Use Recursion# ?5 [+ ]2 {7 {- A
2 l& |8 ^% T3 @6 V Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
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Problems with a clear base case and recursive case.
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# q$ L" Q I3 \/ |Example: Fibonacci Sequence3 V( P/ n5 `9 W, f) d/ `2 q) G2 m4 w
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The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:$ e! M1 ]* V1 c s" }/ ~: J
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Base case: fib(0) = 0, fib(1) = 19 C: @4 c; |; M" V
$ u: ?4 I% g+ w; _/ q: t Recursive case: fib(n) = fib(n-1) + fib(n-2)
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4 w$ p7 E: V- s3 ?python
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def fibonacci(n):' J! Q' Q R8 ^5 K
# Base cases
9 F) b- \7 E( r8 b1 c9 U* m if n == 0:
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elif n == 1:& x6 a3 v1 \- {* F
return 1% k+ w# I. I/ g; \
# Recursive case; y- ~3 d' I* B q/ G7 ?3 ]
else:
9 K" B n# ]5 l return fibonacci(n - 1) + fibonacci(n - 2)
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* r" y0 b5 e3 g# v( f# Example usage
# z9 B/ }- U* ^& Aprint(fibonacci(6)) # Output: 8
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Tail Recursion
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Tail recursion is a special case of recursion where the recursive call is the last operation in the function. Some programming languages optimize tail-recursive functions to avoid stack overflow, but not all languages (e.g., Python does not optimize tail recursion).
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% [6 N" i1 B; K* V6 ?3 _In summary, recursion is a fundamental concept in programming that allows you to solve problems by breaking them into smaller, self-similar subproblems. It’s important to define a base case to avoid infinite recursion and to understand the trade-offs between recursion and iteration. |
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