Chapter 9 Binary Tree and General Tree

Overview ● Two-way decision making is one of the fundamental concepts in computing.  A binary tree models two-way decisions.  A hierarchy represents multi-way choices.  The general tree is an extension of the binary tree.

Learning Objectives ● Describe a binary tree in terms of its structure and components, and learn recursive definitions of the binary tree and its properties. ● Study standard tree traversals in depth. ● Develop a binary tree class interface based on its recursive definition. ● Learn about the signature of a binary tree and understand how to build a binary tree given its signature.

Learning Objectives ● Understand Huffman coding, a binary tree- based text compression application, and use the binary tree class to implement Huffman coding. ● Implement the binary tree class. ● Study how tree traversals may be implemented non-recursively using a stack. ● Describe the properties of a general tree.

Learning Objectives ● Learn the natural correspondence of a general tree with an equivalent binary tree, and the signature of a general tree.

9.1.1 Components

● A binary tree consists of nodes and branches.  A node is a place in the tree where data is stored. ● There is a special node called the root.  “starting point” of the tree. ● The nodes are connected to each other by links or branches.  A left branch or right branch.  Binary means that there are at most two choices. ● A node is said to have at most two children. ● A node that does not have any children is called a leaf. ● Non-leaf nodes are called internal nodes.

9.1.1 Components ● There is a single path from any node to any other node in the tree.

9.1.2 Position as Meaning ● If-then-else tree  Represents an if-then- else construct in a program. ● Every node in this tree is conditional expression that evaluates to yes or no. ● If it evaluates to yes, the left branch (if any) is taken and if evaluates to no, the right branch (if any) is taken.

9.1.2 Position as Meaning ● Expression tree  (f + ((a * b) - c)) Double-click to add graphics

Create Expression tree ● 1- If the current token is a '(', add a new node as the left child of the current node, and descend to the left child. ● 2- If the current token is in the list ['+','-','/','*'], set the root value of the current node to the operator represented by the current token. Add a new node as the right child of the current node and descend to the right child. ● 3- If the current token is a number, set the root value of the current node to the number and return to the parent. ● 4- If the current token is a ')', go to the parent of the current node.

● let’s look at an example of the rules outlined above in action. We will use the expression (3+(4 ∗ 5)). ● We will parse this expression into the following list of character tokens['(', '3', '+', '(', '4', '*', '5',')',')']. ● Initially we will start out with a parse tree that consists of root node.

(3+(4 ∗ 5))

9.1.3 Structure ● Structure  Two trees with the same number of nodes may not have the same structure.

9.1.3 Structure ● Depth is the distance from the root.  Nodes at the same depth are said to be at the same level, with the root being at level zero.  The height of a tree is the maximum level (or depth) at which there is a node.

9.1.3 Structure Full Binary Tree: A binary tree in which all of the leaves are on the same level and every nonleaf node has two children

9.1.3 Structure ● (a), first three are strictly binary, but the fourth is not. first two are FULL binary tree ● (b), first two are complete and the last two are not. ● At level i, there can be at most 2 i nodes.  Maximum number of nodes over all the levels.

9.1.4 Recursive Definitions

Traversal Definitions ● Preorder traversal: Visit the root, visit the left subtree, visit the right subtree ● Inorder traversal: Visit the left subtree, visit the root, visit the right subtree ● Postorder traversal: Visit the left subtree, visit the right subtree, visit the root

Visualizing Binary Tree Traversals

Three Binary Tree Traversals

9.2 Binary Tree Traversals

Binary Search Tree

Overview ● A binary tree possesses ordering property that maintains the data in its nodes in sorted order.  Since the search tree is a linked structure, entries may be inserted and deleted without having to move other entries over, unlike ordered lists in which insertions and deletions require data movement.  The AVL tree is a height-balanced binary search tree that delivers guaranteed worst-case search, insert, and delete times that are all O(log n)

Learning Objectives ● Explore the motivation for binary search trees by learning about the comparison tree for binary search. ● Use the comparison tree as an analytical tool to determine the running time of binary search. ● Describe the binary search tree structure and properties. ● Study the primary binary search tree operations of search, insert, and delete, and analyze their running times.

Learning Objectives ● Understand a binary search tree class interface and use it in application examples. ● Implement the binary search tree class with a binary tree class as the reused storage component. ● Study the AVL tree structure properties, the search, insert, and delete operations, and their running times.

10.2 Binary Search Tree Properties

● All three trees have the same set of keys. ● Their structures are different, depending on the sequence of insertion or deletion.

10.3 Binary Search Tree Operations ● Three foundational operations  Search  Insert  Delete

Search  The tree nodes are for real. ● The target key is compared against the key at the root of the tree. ● If they are equal, sucess. ● If not, recusively search the appropriate child. ● Search terminates with failure if an empty subtree is reached.

Search

Insert ● To insert a value, search must force a failure. ● Item in inserted in the failed location. ● A newly inserted node always becomes a leaf node in the search tree.

Insert

Delete ● The value to be deleted is first located in the binary search tree. ● Three possible cases.  Case a: X is a leaf node.

Delete  Case b: X has one child ● Replace the deleted node with the child.

Delete  Case c: X has two children ● Find the inorder predecessor, Y, of X. ● Copy the entry at Y into X. ● Apply deletion on Y.  Applying deletion on Y will revert to either case b or a since Y is guaranteed to not have a right subtree.

Running Times ● Search: worst case  Tied to the worst possible shape a tree can attain.  Such a tree degenerates into sequential search. ● O(n).

Running Times

● Insertion: worst case  O(n) ● Deletion: worst case  O(n)

Balancing ● Keeping a binary search tree balanced allows the height never to exceed O(log n). ● There are two popular ways of maintaining and constructing balanced binary search trees.  AVL tree.  red-black tree.

10.4 A BinarySearchTree Class

Example Treesort ● inOrder traversal method invokes visitor.visit() when a node is visited.

Example: Treesort

Example: Counting Keys ● Count the number of keys in a binary search tree that are less than a given key. ● It is possible to simply examine every node of the tree.

10.6 BinarySearchTree Class Implementation

Search Implementation

Insert Implementation

● left, right, and parent fields are directly accessed, instead of calling attachLeft and attachRight. ● Clients of BinaryTree that are not in its package are required to use attachLeft or attachRight.

Delete Implementation ● findPredecessor helper method.  Case c requires finding the inorder predeccessor.

Delete Implementation ● deleteHere helper method.

Delete Implementation

Convenience Methods and Traversals ● minValue needs to find the "leftmost" node. ● maxValue finds the "rightmost".

Convenience Methods and Traversals

● This implementation hides the tree structure from its clients. ● All clients need to see are the search, insert, and delete operation. ● "preorder", "inorder", and "postorder" provide a small window into the implementation structure.

10.9 Summary ● A comparison tree for binary search on an array is a binary tree that depicts all possible search paths. ● A failure node in a comparison tree catches a range of values that lie between its inorder predecessor and its in order successor in the tree. ● A comparison tree is simply a conceptual tool to analyze the time taken by binary search, and is therefore often referred to as an implicit search tree.

10.9 Summary ● The shape of the comparison tree is independent of the data entries in the array; it only depends on the length of the array. ● The worst-case number of comparisons for a successful search in 2h-1, and for unsuccessful search is 2h. ● Binary search on an ordered array is O(log n). ● A binary search tree is a binary tree whose entries are arranged in order.

10.9 Summary ● An inorder traversal of a binary search tree will visit the nodes in ascending order of values. ● The values stored in a binary search tree must lend themselves to being arranged in order. ● For any given number of values, n, there is only one binary search comparison tree. There are many binary search trees possible as there are different binary trees that can be constructed out of n nodes.

10.9 Summary ● The worst possible binary tree structure is one that is completely skewed either to the left or right – O(n). ● The worst-case running times for search, insert, and delete in a balanced binary search tree are all O(log n). ● Treesort is an algorithm to sort a set of values by inserting them one by one into a binary search tree, and then visiting them in inorder sequence -- O(n 2 ).

10.9 Summary ● The AVL tree and red-black tree are two of the many types of balanced binary search trees that guarantee a worst case search / insert / delete time of O(log n). ● An AVL tree is a binary search tree in which the heights of the left and right subtrees of every node differ by at most 1. ● Recursive definition: An AVL tree is a binary search tree in which the left and right subtrees of the root are AVL trees whose heights differ by at most 1.

10.9 Summary ● Rotation about a link in an AVL tree takes O(1) time. ● Insertion in an AVL tree starts with a regular binary search tree insertion, followed by rebalancing. ● Deletion in an AVL tree starts with a regular binary search tree deletion, followed by rebalancing.

9.3 A Binary Tree Class

● Delete a single node from a tree.

9.3 A Binary Tree Class ● Recursive traversal procedures.

9.3 A Binary Tree Class ● Running times of methods  makeRoot, setData, and getData involve reading or writing data once.  Checking for whether the pointer is null.  isEmpty O(1).  clear simply have to set the root pointer to null. ● O(1)  attachLeft and attachRight ● Three pointer settings O(1) ● detachLeft and detachRight O(1)

9.3 A Binary Tree Class ● Root called on a leaf node that is at the greatest possible depth in the tree.  O(h) h is the height of the tree.  The height could be as much as n – 1 in the case where the tree is entirely lopsided.  O(n)

9.9 Summary ● Binary trees model two-way decision making systems. ● A binary tree consists of nodes and branches.  There is a special node called the root. ● Every node in a binary tree has at most two children. ● Nodes that have no children are called leaf nodes, others are called internal nodes.

9.9 Summary ● There is a single path between any pair of nodes in a binary tree. ● A binary tree is defined by the relative positions of the data in its nodes, and the tree as a whole carries a meaning that would change if the relative positions of the data in the tree were to change. ● The depth of a node in a binary tree is the number of branches (distance) from the root to that node.

9.9 Summary ● Nodes at the same depth in a binary level are said to be at the same level. ● The height of a binary tree is the maximum level at which there is a node. ● A strictly binary tree is one in which every node has either no child or two children. ● In a complete binary tree, every level but the last must have the maximum number of nodes possible at that level.

9.9 Summary ● The maximum possible number of nodes at level i in a binary tree is 2 i. ● The maximum possible number of nodes in a binary tree of height h is 2 h+1 – 1. ● If N max is the maximum number of nodes in a binary tree, its height is log(N max + 1) – 1. ● Recursive definition: A binary tree is either empty, or it consists of a special node called root that has a left subtree and a right subtree that are mutually disjoint binary trees.

9.9 Summary ● The number of nodes in an empty binary tree is zero.  Otherwise, the number of nodes is one plus the number of nodes each in the left and right subtrees of the root. ● The height of an empty binary tree is -1.  Otherwise, the height is one plus the maximum of the heights of the left and right subtrees of the root.

9.9 Summary ● Recursive definition of inorder traversal type T: first recursively traverse the Left subtree of T, then Visit the root of T, then recursively traverse the Right subtree of T. ● Recursive definition of preorder traversal of tree T: first Visit the root of T, then recursively traverse the Left subtree of T, then recursively traverse the Right subtree of T.

9.9 Summary ● Recursive definition of postorder traversal of treeT: first recursively traverse the Left subtree of T, then recursively traverse the Right subtree of T, then Visit the root of T. ● The recursive traversals may be written in short form as follows: inorder is LVR, preorder is VLR, and postorder is LRV. ● Level-order traversal of treeT: starting at the root level, go level by level in T, visiting the nodes at any level in left to right order.