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A tree
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Figure 89 illustrates a tree The root node is A: A's children are B, C, D, and E Because A is the root, it has no parent; all other nodes have parents For instance, B's parent is A A node that has no children is called leaf The leaves in this tree are C, E G, H, I, and K The length of the path from A to K is 3 (edges); the length of the path from A to A is 0 (edges)
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A leaf has no children
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Perhaps the best way to understand recursion is to consider examples In this section, we look at four more examples of recursion The first two are easily implemented nonrecursively, but the last two show off some of the power of recursion The last two examples draw recursive pictures; unfortunately, C++ does not have a standard way of doing graphics Consequently the code for those examples is in Java The code is identical, syntactically, to C++ code, and thus the Java syntax should not get in the way of understanding the examples
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1 / / Evaluate n! 2 long factorial( int n
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if( n < = 1 ) / / base case return 1; else return n * factorial( n - 1
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Recursive implementation of the factorial function
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Recall that N! is the product of the first N integers Thus we can express N! as N times ( N - l ) ! Combined with the base case l ! = 1, this information immediately provides all that we need for a recursive implementation It is shown in Figure 810
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In Section 662 we described the binary search Recall that in a binary search, we perform a search in a sorted array A by examining the middle element If we have a match, we are done Otherwise, if the item being searched for is smaller than the middle element, we search in the subarray that is to the left of the middle element Otherwise, we search in the subarray that is to the right of the middle element This procedure presumes that the subarray is not empty; if it is, the item is not found This description translates directly into the recursive method shown in Figure 811 The code illustrates a thematic technique in which the public driver routine makes an initial call to a recursive routine and passes on the return value Here, the driver sets the low and high points of the subarray, namely, 0 and asize( )-I In the recursive method, the base case at lines 15 and 16 handles an empty subarray Otherwise, we follow the description given previously by making a recursive call on the appropriate subarray (line 21 or 23) if a match has not been detected When a match is detected, the matching index is returned at line 25 Note that the running time, in terms of Big-Oh, is unchanged from the nonrecursive implementation because we are performing the same work In practice, the running time would be expected to be slightly larger because of the hidden costs of recursion
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
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Performs the standard binary search using two comparisons per level This is a driver that calls the recursive method template <class Comparable> int binarySearch( const vector<Comparable> & a, const Comparable & x )
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return binarysearch ( a, x, 0 , asize (
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/ / Recursive routine template <class Comparable> int bindrysearch( const vector<Comparable> & a, const Comparable & x, int low, int high
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if( low > high ) return NOT-FOUND; int mid
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high
if( a[ mid return else if ( x return else return
I < x ) bindrysearch( a, x, mid + 1, high ) ; < a[ mid I ) bindrysearch( a, x, low, mid - 1 ) ;
mid;