High School Statutory Authority: Reading, where students read and understand a wide variety of literary and informational texts; Writing, where students compose a variety of written texts with a clear controlling idea, coherent organization, and sufficient detail; Research, where students are expected to know how to locate a range of relevant sources and evaluate, synthesize, and present ideas and information; Listening and Speaking, where students listen and respond to the ideas of others while contributing their own ideas in conversations and in groups; and Oral and Written Conventions, where students learn how to use the oral and written conventions of the English language in speaking and writing. The standards are cumulative--students will continue to address earlier standards as needed while they attend to standards for their grade. In English I, students will engage in activities that build on their prior knowledge and skills in order to strengthen their reading, writing, and oral language skills.
The trick is to pick a midpoint near the center of the array, compare the data at that point with the data being searched and then responding to one of three possible conditions: Recursion is used in this algorithm because with each pass a new array is created by cutting the old one in half.
The binary search procedure is then called recursively, this time on the new and smaller array. Typically the array's size is adjusted by manipulating a beginning and ending index.
The algorithm exhibits a logarithmic order of growth because it essentially divides the problem domain in half with each pass. Example implementation of binary search in C: Recursive data type An important application of recursion in computer science is in defining dynamic data structures such as lists and trees.
Recursive data structures can dynamically grow to a theoretically infinite size in response to runtime requirements; in contrast, the size of a static array must be set at compile time.
This term refers to the fact that the recursive procedures are acting on data that is defined recursively. As long as a programmer derives the template from a data definition, functions employ structural recursion. That is, the recursions in a function's body consume some immediate piece of a given compound value.
Linked list Below is a C definition of a linked list node structure. Notice especially how the node is defined in terms of itself. The "next" element of struct node is a pointer to another struct node, effectively creating a list type. For each node it prints the data element an integer.
Binary tree Below is a simple definition for a binary tree node.
Like the node for linked lists, it is defined in terms of itself, recursively. There are two self-referential pointers: Note that because there are two self-referencing pointers left and righttree operations may require two recursive calls: A Binary search tree is a special case of the binary tree where the data elements of each node are in order.
Filesystem traversal[ edit ] Since the number of files in a filesystem may vary, recursion is the only practical way to traverse and thus enumerate its contents.
Traversing a filesystem is very similar to that of tree traversaltherefore the concepts behind tree traversal are applicable to traversing a filesystem.
More specifically, the code below would be an example of a preorder traversal of a filesystem. It is, essentially, a recursive implementation, which is the best way to traverse a filesystem.
It is also an example of direct and indirect recursion. The method "rtraverse" is purely a direct example; the method "traverse" is the indirect, which calls "rtraverse.
Implementation issues[ edit ] In actual implementation, rather than a pure recursive function single check for base case, otherwise recursive stepa number of modifications may be made, for purposes of clarity or efficiency.
Wrapper function at top Short-circuiting the base case, aka "Arm's-length recursion" at bottom Hybrid algorithm at bottom — switching to a different algorithm once data is small enough On the basis of elegance, wrapper functions are generally approved, while short-circuiting the base case is frowned upon, particularly in academia.4 Writing Structured Programs.
By now you will have a sense of the capabilities of the Python programming language for processing natural language. The Fibonacci sequence, and recursive sequences generally, never fail to reward our study with surprising and intriguing connections between mathematics and the natural world.
Directions: These questions will ask you to write formulas for sequences. While one solution is offered, be aware that there are other notations and forms that may also be acceptable. While one solution is offered, be aware that there are other notations and forms that may also be acceptable.
A recursive formula always uses the preceding term to define the next term of the sequence. Sequences can have the same formula but because they start with a different number, they are different patterns.
Acknowledgements. The contributions to early versions of this manual by Saikat DebRoy (who wrote the first draft of a guide to ashio-midori.com ashio-midori.comal) and Adrian Trapletti (who provided information on the C++ interface) are gratefully acknowledged. The Fibonacci sequence, and recursive sequences generally, never fail to reward our study with surprising and intriguing connections between mathematics and the natural world.