CS 206 Introduction to Computer Science II
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1 CS 206 Introduction to Computer Science II 07 / 15 / 2016 Instructor: Michael Eckmann
2 Today s Topics Questions? Comments? Binary trees implementation Binary search trees Michael Eckmann - Skidmore College - CS Summer 2016
3 Implementation of binary trees Similar to how we implemented a linked list of nodes, we can implement a binary tree of nodes where each node contains a reference to a left child and a right child (each of which could be null.) public class BTNode { private int data; public BTNode left; public BTNode right; } // constructor that sets left and right to null public BTNode(int i) { data = i; left = null; right = null; } Michael Eckmann - Skidmore College - CS Summer 2016
4 Implementation of binary trees public class BinaryTree { private BTNode root; } // constructor that sets root to null public BinaryTree() { root = null; } // plenty more methods here to insert nodes, etc... Michael Eckmann - Skidmore College - CS Summer 2016
5 A few operations on binary trees getleftmost node Starting at root, follow the left until you hit a node whose left is null. That node is the leftmost node. getrightmost node Starting at root, follow the right until you hit a node whose right is null. That node is the rightmost node. According to these definitions, will the leftmost and rightmost nodes always be leaves? Michael Eckmann - Skidmore College - CS Summer 2016
6 A few operations on binary trees The leftmost node can have a right child and the rightmost node can have a left child, so the leftmost and rightmost nodes in a binary tree aren't necessarily leaves. Later we'll talk about how to create recursive methods to remove these nodes. Michael Eckmann - Skidmore College - CS Summer 2016
7 Traversals of binary trees There are three typical ways to traverse a binary tree preorder, postorder and inorder. preorder process root process nodes in left subtree with a recursive call process nodes in right subtree with a recursive call Example on the board of a preorder traversal. Michael Eckmann - Skidmore College - CS Summer 2016
8 Traversals of binary trees postorder process nodes in left subtree with a recursive call process nodes in right subtree with a recursive call process root Example on the board of a postorder traversal. Michael Eckmann - Skidmore College - CS Summer 2016
9 Traversals of binary trees inorder process nodes in left subtree with a recursive call process root process nodes in right subtree with a recursive call Example on the board of an inorder traversal. Applet: Michael Eckmann - Skidmore College - CS Summer 2016
10 Implementing the traversals of binary trees Let's assume the processing we're doing to each node is just printing the data in that node. So, for preorder traversal we need to do the following print the root's data do a preorder traversal of the left subtree do a preorder traversal of the right subtree Notice the recursion? Michael Eckmann - Skidmore College - CS Summer 2016
11 Implementing the traversals of binary trees The only issue is when to stop the recursion. To do a preorder traversal of the left subtree there has to be a left subtree. If there is no left subtree (that is, left == null) then don't traverse that anymore. Same for a preorder traversal of the right subtree - there has to be a right subtree to traverse. If there is no right subtree (that is, right == null) then don't traverse that anymore. Michael Eckmann - Skidmore College - CS Summer 2016
12 Implementing the traversals of binary trees So, for preorder traversal we need to do the following print the root's data if (left!= null) do a preorder traversal of the left subtree if (right!= null) do a preorder traversal of the right subtree Michael Eckmann - Skidmore College - CS Summer 2016
13 Implementing binary tree traversals // this method lives inside the BinaryTree class and can only be called // from within it private void preorderprint(btnode node) { } System.out.println(node.data); if (node.left!= null) preorderprint(node.left); if (node.right!= null) preorderprint(node.right); Michael Eckmann - Skidmore College - CS Summer 2016
14 Implementing binary tree traversals // we need a way to start the traversal off... // this method lives inside the BinaryTree class public void preorderprint() { } if (root!= null) preorderprint(root); Michael Eckmann - Skidmore College - CS Summer 2016
15 binary tree representation Besides representing a binary tree as a structure of nodes where each node has data and references to other nodes (left and right). we can store a binary tree in an array. The 0 th index will hold the root data, the 1 st and 2 nd will hold the root's left and right children's data respectively. The root's left child's left and right children's data are in 3 rd and 4 th indices etc. Each level of the binary tree is stored in contiguous indices. Michael Eckmann - Skidmore College - CS Summer 2016
16 binary tree representation Node i's children are at 2i + 1 and 2i + 2. Example: root is at 0, it's left child is at 1 and right is at 2. i=0, 2i+1 = 1, 2i+2 = 2 Another example: the root's right child is at 2 and that node's children are at 5 and 6. i=2, 2i+1 = 5, 2i+2 = 6 Now, given an index of a node, how can we determine where the parent lives? Michael Eckmann - Skidmore College - CS Summer 2016
17 binary tree representation This scheme only works really well for perfect binary trees or at least complete binary trees. Why? Because if we know the number of nodes in the tree, that is the number of elements in the array. Problem when we want to add to the tree. Possible solution is to create a really large array and keep track of the index of the deepest, right node and never allow any code to look at any indices higher than that except when adding nodes. Problem if the tree is not complete. We'll have holes in the array. Michael Eckmann - Skidmore College - CS Summer 2016
18 binary tree representation Problem if the tree is not complete. We'll have holes in the array. Why is this a problem? What possible solutions are there to storing an incomplete binary tree in an array? Michael Eckmann - Skidmore College - CS Summer 2016
19 binary tree representation What possible solutions are there to storing an incomplete binary tree in an array? Could keep another array of booleans whose elements are true if the index holds a node, false if the index does not hold a node. Or if the binary tree array is an array of references, ones that do not hold a node can be null. Michael Eckmann - Skidmore College - CS Summer 2016
20 Abstract data types Now is a good time to bring up the term abstract data type (ADT). We just discussed a few different ways of storing / representing a binary tree. But these different representations were assumed to allow the same exact operations, such as traversing the tree in some order, or finding the children of a node, or finding all the ancestors of a node, etc. So, to a user of the binary tree data type, the implementation doesn't matter. Only the interface to the data type is needed to be known (e.g. What methods can be called and the parameters they take). How the data type was implemented is not needed to be known by the user of the data type. Michael Eckmann - Skidmore College - CS Summer 2016
21 Abstract data types This is a kind of information hiding. Hide how it's implemented, while still providing a way to use it. For example: The first programming assignment required a way to store a list of the students' names and scores. Different implementations were with a linked list, an Arraylist, and an array. Ideally the class containing your main method shouldn't have had to know anything about which way the list was implemented. It should have just needed to use the list by calling addstudent, changescore etc. Michael Eckmann - Skidmore College - CS Summer 2016
22 Abstract data types Information hiding is a good thing to keep in mind while coding. Think about what operations/methods are needed to be provided for users of your class and provide them, but hide everything else. This will help you to keep your classes separated well. Anyone have any comments? Michael Eckmann - Skidmore College - CS Summer 2016
23 Binary Search Trees Binary Search Trees (BSTs) Definition Example of a BST and verification that it is one Are BST's unique for a given set of data? Algorithms print keys in ascending order Search for a key Find minimum key Find maximum key Insert a key
24 Binary Search Trees Binary Search Trees (BSTs) A tree that is both a binary tree and a search tree. we know the definition of a tree and a binary tree so: We just need to define search tree. A search tree is a tree where every subtree of a node has data (aka keys) less than any other subtree of the node to its right. the keys in a node are conceptually between subtrees and are greater than any keys in subtrees to its left and less than any keys in subtrees to its right.
25 Binary Search Trees A binary search tree is a tree that is a binary tree and is a search tree. In other words it is a tree that has at most two children for each node and every node's left subtree has keys less than the node's key, and every right subtree has keys greater than the node's key. Definitions taken from (The National Institute of Standards and Technology)
26 Binary Search Trees Each node in a BST has key left reference right reference parent reference
27 Binary Search Trees F / \ B H / \ \ A D K Assume Alphabet ordering of letters. Let's verify whether it is indeed a binary search tree. What properties does it need to have again? Does it satisfy those properties?
28 Binary Search Trees Are BST's unique for a given set of keys? Let's build a tree for the list of keys 13, 45, 10, 9, 54, 11, 42 Choose 13 to put in the root and go from there
29 Binary Search Trees Are BST's unique for a given set of keys? Let's build a tree for the list of keys 13, 45, 10, 9, 54, 11, 42 What if we change the order that we insert the keys? What if we choose a different key as the root to start?
30 Binary Search Trees Let's look at algorithms to do the following Print keys in ascending order Search for a key Find minimum key Find maximum key Insert a key Delete a key Height of a BST
31 Print the Keys To print the keys in increasing order we use inorder traversal. Recursive description of inorder traversal In-order traversal of a tree Start with x being the root check if x is not null then 1) In-order traversal of left(x) 2) print key(x) 3) In-order traversal of right(x) Let's apply this algorithm to the tree and see what it does. What is the running time complexity of this algorithm for a tree that has n nodes?
32 Search for a key To search in a binary search tree for a key k, start with x being the root. Here's an iterative solution of the search public BSTNode searchkey(int searchkey) { BSTNode temp = root; } while (temp!= null && temp.key!= searchkey) { if (temp.key > searchkey) temp = temp.left; else temp = temp.right; } return temp; // may be null or not (if null then key wasn't found) What's the running time of this?
33 Search for a key To search in a binary search tree for a key k, start with x being the root. Here's a recursive solution of the search public BSTNode treesearch(bstnode temp, int k) { if (temp == null k == temp.key) return temp; if (k < temp.key) return treesearch(temp.left, k) else return treesearch(temp.right, k) } What's the running time of this?
34 Search for a key What's the running time of this? On the order of the height of the tree. What if the binary search tree is complete (or full)?
35 Find Minimum key in a BST How might we devise an algorithm to find the minimum? Where in the tree is the minimum value?
36 Find Minimum key in a BST How might we devise an algorithm to find the minimum? Where in the tree is the minimum value? It is in the leftmost node while (temp.left!= null) temp = temp.left; return temp;
37 Find Maximum key in a BST How might we devise an algorithm to find the maximum? Where in the tree is the maximum value?
38 Find Maximum key in a BST How might we devise an algorithm to do find the maximum? Where in the tree is the maximum value? It is in the rightmost node while (temp.right!= null) temp = temp.right; return temp; Running times of these?
39 Insert a key in a BST How might we devise an algorithm to insert a key into the tree? Can the key go anywhere?
40 Insert a key in a BST How might we devise an algorithm to insert a key into the tree? Can the key go anywhere? No, it has to follow the rules of BST's so the resulting tree after insert must be a BST. z is the node to insert and z.key is its key and its left, right and parents are null. Need to keep track of where we are in the tree as we traverse it and the parent of where we are because we might have to go back up the tree. par will be a pointer to the parent of temp as we go through the tree
41 Insert a key in a BST insert(bstnode z) // insert node z in tree T { BSTNode temp = root; BSTNode par = null; // search the tree to find the place it should go while (temp!= null) { } par = temp; if (z.key < temp.key) temp = temp.left; else temp = temp.right; // continued on next slide
42 Insert a key in a BST // now we know x is null and the parent of x is par so, z.parent = par; // insert the key either at the root or to the left or right of par if (par = null) root = z; else if (z.key < par.key) else par.left = z; par.right = z; } // any fear of overwriting a node? What if par.left or par.right // contain a node?
43 Insert a key in a BST // any fear of overwriting a node? What if par.left or par.right // contain a node? They can't contain a node because of the first half of the algorithm guarantees it Examples: Let's insert C into the original tree. Then let's insert E.
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