Binary Tree and General Tree

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Chapter 9

• Binary Tree and General Tree

2
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.

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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.

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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.

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Learning Objectives

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

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9.1.1 Components
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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.

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9.1.1 Components

• There is a single path from any node to any other
  node in the tree.

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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.

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9.1.2 Position as Meaning

• Expression tree
• (f ((a b) - c))

Double-click to add graphics
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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.

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• lets look at an example of the rules outlined
  above in action. We will use the
  expression (3(45)).
• 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.

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(3(45))
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9.1.3 Structure

• Structure
• Two trees with the same number of nodes may not
  have the same structure.

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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.

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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

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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 2i nodes.
• Maximum number of nodes over all the levels.

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9.1.4 Recursive Definitions
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9.1.4 Recursive Definitions
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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

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Visualizing Binary Tree Traversals
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Three Binary Tree Traversals
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9.2 Binary Tree Traversals
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9.2 Binary Tree Traversals
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• Binary Search Tree

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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)

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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.

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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.

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10.2 Binary Search Tree Properties
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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.

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10.3 Binary Search Tree Operations

• Three foundational operations
• Search
• Insert
• Delete

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10.3.1 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.

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10.3.1 Search
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10.3.2 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.

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10.3.2 Insert
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10.3.3 Delete

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

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10.3.3 Delete

• Case b X has one child
• Replace the deleted node with the child.

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10.3.3 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.

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Running Times

• Search worst case
• Tied to the worst possible shape a tree can
  attain.
• Such a tree degenerates into sequential search.
• O(n).

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Running Times
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Running Times

• Insertion worst case
• O(n)
• Deletion worst case
• O(n)

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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.

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10.4 A BinarySearchTree Class
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10.4 A BinarySearchTree Class
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10.5.1 Example Treesort

• inOrder traversal method invokes visitor.visit()
  when a node is visited.

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10.5.1 Example Treesort
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10.5.2 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.

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10.6 BinarySearchTree Class Implementation
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10.6 BinarySearchTree Class Implementation
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10.6.1 Search Implementation
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10.6.2 Insert Implementation
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10.6.2 Insert Implementation
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10.6.2 Insert Implementation
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10.6.2 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.

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10.6.3 Delete Implementation

• findPredecessor helper method.
• Case c requires finding the inorder predeccessor.

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10.6.3 Delete Implementation

• deleteHere helper method.

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10.6.3 Delete Implementation
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10.6.3 Delete Implementation
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10.6.3 Delete Implementation
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10.6.4 Convenience Methods and Traversals

• minValue needs to find the "leftmost" node.
• maxValue finds the "rightmost".

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10.6.4 Convenience Methods and Traversals
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10.6.4 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.

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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.

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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.

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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.

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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(n2).

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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.

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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.

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9.3 A Binary Tree Class
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9.3 A Binary Tree Class
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9.3 A Binary Tree Class

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9.3 A Binary Tree Class
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9.3 A Binary Tree Class

• Delete a single node from a tree.

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9.3 A Binary Tree Class

• Recursive traversal procedures.

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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)

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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)

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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.

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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.

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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.

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9.9 Summary

• The maximum possible number of nodes at level i
  in a binary tree is 2i.
• The maximum possible number of nodes in a binary
  tree of height h is 2h1 1.
• If Nmax is the maximum number of nodes in a
  binary tree, its height is log(Nmax 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.

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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.

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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.

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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.