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• In order to understand recursion, you must

• first understand recursion.

• Having recursion in program design means that you have self-referential definitions.

• Recursive data structures, for instance, are data structures that include themselves in their definitions.

• But today, we're going to focus on recursive functions.

• Recall that functions take inputs, arguments, and return a value as their output represented by this diagram here.

• We'll think of the box as the body of the function,

• containing the set of instructions that interpret the input and provide an output.

• Taking a closer look inside the body of the function could reveal calls to other functions as well.

• Take this simple function, foo, that takes a single string as input and

• prints how many letters that string has.

• The function strlen, for string length, is called, whose output is

• required for the call to printf.

• Now, what makes a recursive function special is that it calls itself.

• We can represent this recursive call with this orange arrow looping back to itself.

• But executing itself again will only make another recursive call, and

• another and another.

• But recursive functions can't be infinite.

• They have to end somehow, or your program will run forever.

• So we need to find a way to break out of the recursive calls.

• We call this the base case.

• When the base case condition is met, the function returns without making another recursive call.

• Take this function, hi, a void function that takes an int n as input.

• The base case comes first.

• If n is less than zero, print bye and return For all other cases, the

• function will print hi and execute the recursive call.

• Another call to the function hi with a decremented input value.

• Now, even though we print hi, the function won't terminate until we

• return its return type, in this case void.

• So for every n other than the base case, this function hi will return hi of n minus 1.

• Since this function is void though, we won't explicitly type return here.

• We'll just execute the function.

• So calling hi(3) will print hi and execute hi(2) which executes hi(1) one

• which executes hi(0), where the base case condition is met.

• So hi(0) prints bye and returns.

• OK.

• So now that we understand the basics of recursive functions, that they need

• at least one base case as well as a recursive call, let's move on to a

• more meaningful example.

• One that doesn't just return void no matter what.

• Let's take a look at the factorial operation used most commonly in probability calculations.

• Factorial of n is the product of every positive integer less than and equal to n.

• So factorial five is 5 times 4 times 3 times 2 times 1 to give 120.

• Four factorial is 4 times 3 times 2 times 1 to give 24.

• And the same rule applies to any positive integer.

• So how might we write a recursive function that calculates the factorial of a number?

• Well, we'll need to identify both the base case and the recursive call.

• The recursive call will be the same for all cases except for the base

• case, which means that we'll have to find a pattern that will give us our desired result.

• For this example, see how 5 factorial involves multiplying 4 by 3 by 2 by 1

• And that very same multiplication is found here, the

• definition of 4 factorial.

• So we see that 5 factorial is just 5 times 4 factorial.

• Now does this pattern apply to 4 factorial as well?

• Yes

• We see that 4 factorial contains the multiplication 3 times 2 times 1, the

• very same definition as 3 factorial.

• So 4 factorial is equal to 4 times 3 factorial, and so on and so forth our

• pattern sticks until 1 factorial, which by definition is equal to 1.

• There's no other positive integers left.

• So we have the pattern for our recursive call.

• n factorial is equal to n times the factorial of n minus 1.

• And our base case?

• That'll just be our definition of 1 factorial.

• So now we can move on to writing code for the function.

• For the base case, we'll have the condition n equals equals 1, where

• we'll return 1.

• Then moving onto the recursive call, we'll return n times the

• factorial of n minus 1.

• Now let's test this our.

• Let's try factorial 4.

• Per our function it's equal to 4 times factorial(3).

• Factorial(3) is equal to 3 times factorial(2).

• Factorial(2) is equal to 2 times factorial(1), which returns 1.

• Factorial(2) now returns 2 times 1, 2.

• Factorial(3) can now return 3 times 2, 6.

• And finally, factorial(4) returns 4 times 6, 24.

• If you're encountering any difficulty with the recursive call, pretend that

• What I mean by this is that you should trust your recursive calls to return

• the right values.

• For instance, if I know that factorial(5) equals 5 times

• factorial(4), I'm going to trust that factorial(4) will give me 24.

• Think of it as a variable, if you will, as if you already defined factorial(4).

• So for any factorial(n), it's the product of n and the previous factorial.

• And this previous factorial is obtained by calling factorial of n minus 1.

• Now, see if you can implement a recursive function yourself.

• Load up your terminal, or run.cs50.net, and write a function sum

• that takes an integer n and returns the sum of all consecutive positive integers from n to 1.

• I've written out the sums of some values to help you our.

• First, figure out the base case condition.

• Then, look at sum(5).

• Can you express it in terms of another sum?

• How can you express sum(4) in terms of another sum?

• Once you have sum(5) and sum(4) expressed in terms of other sums, see

• if you can identify a pattern for sum(n).

• If not, try a few other numbers and express their sums in terms of another numbers.

• By identifying patterns for discrete numbers, you're well on your way to identifying the pattern for any n.

• Recursion's a really powerful tool, so of course it's not limited to

• mathematical functions.

• Recursion can be used very effectively when dealing with trees for instance.

• Check out the short on trees for a more thorough review, but for now

• recall that binary search trees, in particular, are made up of nodes, each

• with a value and two node pointers.

• Usually, this is represented by the parent node having one line pointing

• to the left child node and one to the right child node.

• The structure of a binary search tree lends itself well

• to a recursive search.

• The recursive call either passes in the left or the right node, but more of that in the tree short.

• Say you want to perform an operation on every single node in a binary tree.

• How might you go about that?

• Well, you could write a recursive function that performs the operation

• on the parent node and makes a recursive call to the same function, passing in the left and right child nodes.

• For example, this function, foo, that changes the value of a given node and

• all of its children to 1.

• The base case of a null node causes the function to return, indicating

• that there aren't any nodes left in that sub-tree.

• Let's walk through it.

• The first parent is 13.

• We change the value to 1, and then call our function on the left as well as the right.

• The function, foo, is called on the left sub-tree first, so the left node

• will be reassigned to 1 and foo will be called on that node's children,

• first the left and then the right, and so on and so forth.

• And tell them that branches don't have any more children so the same process

• will continue for the right children until the whole tree's nodes are reassigned to 1.

• As you can see, you aren't limited to just one recursive call.

• Just as many as will get the job done.

• What if you had a tree where each node had three children,

• Left, middle, and right?

• How would you edit foo?

• Well, simple.

• Just add another recursive call and pass in the middle node pointer.

• Recursion is very powerful and not to mention elegant, but it can be a

• difficult concept at first, so be patient and take your time.

• It's usually the easiest to identify, and then you can work

• backwards from there.

• You know you need to reach your base case, so that might give you a few hints.

• Try to express one specific case in terms of other cases, or in sub-sets.

• Thanks for watching this short.

• At the very least, now you can understand jokes like this.

• My name is Zamyla, and this is cs50.

• Take this function, hi, a void function that takes an int, n, as input.

• The base case comes first.

• If n is less than 0, print "bye" and return.

In order to understand recursion, you must

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# Recursion

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Jjli Li posted on 2016/07/06
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