How do hashes work? Sunday, December 05, :28 PM

Transcription

1 Trees Page 1 How do hashes work? 5:28 PM How do Perl hashes work? So far, we've found out that Perl hashes can do most anything. But how do they work? Key to how hashes work: associating a key with a value.

2 Trees Page 2 Trees 5:22 PM Trees A simplification of graphs No cycles Undirected edges. Rooted trees A specific node (vertex) is the root. (could in theory be any node). Why trees are important: Represent divide-and-conquer processes. Represent decisions/probabilities. Enable efficient searching. Easier to manipulate than graphs.

3 Trees Page 3 Some examples of trees 5:24 PM

4 Trees Page 4 Why trees are important 5:32 PM Why trees are important: Represent divide-and-conquer processes. Represent decisions/probabilities. Enable efficient searching. Easier to manipulate than graphs.

5 Trees Page 5 Kinds of trees 5:31 PM Decision trees: represent decisions in a process Search trees: store data for easy retrieval. Syntax trees: describe how a computer language works.

6 Trees Page 6 Decision trees 5:33 PM Decision trees Nodes represent decisions. Children are later decisions or outcomes. Edges may be labeled with probabilities.

7 Trees Page 7 Example of a decision tree 5:37 PM

8 Trees Page 8 Expression trees 5:34 PM Expression trees Depict the order in which expressions are computed. One way to produce an expression tree: recursive-descent parsing Start with whole expression. Choose the operator with least precedence. Split the expression into three parts: The left-hand side of the operator expression (left child). The operator (root) The right-hand side of the operator expression (right child). Repeat for children!

9 Trees Page 9 Example of an expression tree 5:37 PM

10 Trees Page 10 Binary search trees. 5:24 PM Binary search trees Can store key/value pairs for quick retrieval By divide-and-conquer. In O(log n) time (n=# of pairs) Tree structure: Every node has at most two children. Every node corresponds to a search key and value. For every node in the tree: The left child's key is less than the node's key. The right child's key is greater than the node's key.

11 Trees Page 11 Searching a binary search tree 5:40 PM Searching a binary search tree Input: a binary search tree and a desired key Output: the value associated with that key. Let the current node be the root. If the current node's key is the desired key, return its value. If the current node's key is less than the desired key, go to the right child. If the current node's key is greater than the desired key, go to the left child. Repeat until you find the key, or cannot go in the appropriate direction.

12 Trees Page 12 Example of binary search 5:45 PM

13 Trees Page 13 Building a binary search tree 5:45 PM Building a binary search tree Search for the key. If you find it, stop. Whenever you get to a place where you need to go left or right and can't, put the new key there.

14 Trees Page 14 Example of building a binary search tree 5:47 PM

15 Trees Page 15 Balance 5:47 PM Q: How fast is searching a binary search tree? A: Depends upon how balanced the tree is. This depends upon how the tree is built.

16 Trees Page 16 Same data, balanced and unbalanced. 5:48 PM

17 Trees Page 17 Traversals 5:48 PM Often, we want to do something to every node of a tree. Example: print every search key in a binary search tree. Example: print the original expression that is represented by an expression tree.

18 Trees Page 18 Three kinds of traversals 5:49 PM Three kinds of traversals Inorder: do something to the left tree, then the right tree, then the root. Preorder: do something to the root, then to the left subtree, then to the right subtree. Postorder: do something to the left and right subtrees, then to the root.

19 Trees Page 19 Examples of inorder traversal 5:51 PM

20 Trees Page 20 Some interesting facts 5:52 PM Inorder traversal of an expression tree visits the nodes in their order in the original expression. Inorder traversal of a binary search tree visits the nodes in sorted order.

21 Trees Page 21 Trees in Perl 5:55 PM Trees in Perl Two representations: As nested arrays. As nested hashes. Let's use the simpler form for now.

22 Trees Page 22 Implementing binary search trees 6:17 PM Implementing binary search trees Can use recursion or iteration. Let's try both.