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CS 206 Introduction to Computer Science II 09 / 22 / 2008 Instructor: Michael Eckmann

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 Today’s Topics Questions? Comments? Linked lists –Last few comments about doubly linked lists Trees –Binary trees –Terminology –Binary tree example application –getting leftmost, rightmost nodes –preOrder, inOrder, postOrder traversals using recursion

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 Tree terminology definitions A tree is a data structure consisting of a finite (possibly empty) set of nodes. If the tree is nonempty it has one root node. –All nodes in a tree can have 0 or more children. –All nodes in a tree have exactly one parent, except the root which has 0 parents. –Starting at any node in the tree you can trace a path back to the root by following the parent at each step. Picture on the board.

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 Tree terminology definitions A binary tree is a data structure consisting of a finite (possibly empty) set of nodes. If the tree is nonempty it has one root node. –All nodes in a tree can have 0, 1 or 2 children. –All nodes in a tree have exactly one parent, except the root which has 0 parents. –Starting at any node in the tree you can trace a path back to the root by following the parent at each step. Picture on the board.

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 Tree terminology definitions The root node in a tree is the one without a parent. The parent of a node n, in a tree, is the node that has n as one of its children. Leaf nodes are those that have 0 children. A subtree is part of a tree that has as its root any node in the original tree. The depth of a node is the number of steps away that node is from the root. The depth of the root is 0. The depth of the root's children are 1, etc. The depth of a tree is the maximum depth of any of its leaves.

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 Tree terminology definitions (corrected) A full binary tree is a binary tree where node has either 0 or 2 children. A perfect binary tree is a binary tree where every leaf has the same depth AND all internal nodes have degree 2. A complete binary tree is a binary tree where the leaves with the maximum depth are all on the left (and any other leaves are only one depth less). In other words, every level of the tree except the deepest level, must contain as many nodes as possible. At the deepest level the nodes must be as far left as possible.

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 Tree terminology definitions Two nodes are siblings if they have the same parent. The ancestors of a node are all parents at each step up the tree to the root. The first ancestor of a node is it's parent. The second ancestor of a node is it's parent's parent. And so on until you reach the root which is an ancestor of every node in the tree. The descendents of a node are all it's children and their children's children etc.

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 Exercise How many nodes are there in a full binary tree of depth 8?

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 Exercise How many nodes are there in a full binary tree of depth 8? –root level has 1 node (which is 2 0 ) –next level has 2 nodes (which is 2 1 ) –next level has 4 nodes (which is 2 2 ) –next level has 8 nodes (which is 2 3 ) –next level has 16 nodes (which is 2 4 ) –next level has 32 nodes (which is 2 5 ) –next level has 64 nodes (which is 2 6 ) –next level has 128 nodes (which is 2 7 ) –next level has 256 nodes (which is 2 8 ) These added up are 2 9 – 1 = 512 – 1 = 511.

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 Exercise How many nodes are there in a complete binary tree of depth 4?

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 Exercise How many nodes are there in a complete binary tree of depth 4? –root level has 1 node (which is 2 0 ) –next level has 2 nodes (which is 2 1 ) –next level has 4 nodes (which is 2 2 ) –next level has 8 nodes (which is 2 3 ) –next level has anywhere from 1 to 16 nodes So, adding these is anywhere from 16 to 31 nodes. When a binary tree has 31 nodes is it guaranteed to be a full binary tree?

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 Exercise When a binary tree has 2 n – 1, where n is an integer >= 0, nodes and all nodes have either 0 or 2 children and all leaves are at the same depth is it guaranteed to be a full binary tree? Is it guaranteed to be a complete binary tree?

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 Exercise When a binary tree has 2 n – 1, where n is an integer >= 0, nodes and all nodes have either 0 or 2 children and all leaves are at the same depth is it guaranteed to be a full binary tree? –Yes Is it guaranteed to be a complete binary tree? –Yes, all full binary trees are complete Is a binary tree of 12 nodes –full? –complete?

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 Exercise Is a binary tree of 12 nodes –full? No –complete? possibly

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 Application of binary trees The game 20 questions –The game consists only of yes/no questions Each node can contain a question (or answer) If it contains a question, –If the answer to the question is yes then the left child node contains the next question to ask (or the answer to give) –If the answer to the question is no then the right child node contains the next question to ask (or the answer to give)

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 Application of binary trees What could you say about the nodes that contain answers, not questions? Assuming a full binary tree of depth 20 to be used for the game, how many possible answers could your tree differentiate between?

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 Application of binary trees What could you say about the nodes that contain answers, not questions? –They are leaves. Assuming a full binary tree of depth 20 to be used for the game, how many possible answers could your tree differentiate between? 2 20 = 1048576 (over 1 million) that's the number of leaves The number of questions in the tree are 2 20 – 1 = 1048575 The number of nodes in the tree are 2 21 – 1 = 2097151

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 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?

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 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.

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 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.

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 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.

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 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: http://www.cosc.canterbury.ac.nz/mukundan/dsal/BTree.html

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 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?

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 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.

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Michael Eckmann - Skidmore College - CS 206 - Fall 2008 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

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