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Fundamentals of Python: From First Programs Through Data Structures. Chapter 18 Hierarchical Collections: Trees. Objectives. After completing this chapter, you will be able to: Describe the difference between trees and other types of collections using the relevant terminology

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fundamentals of python from first programs through data structures

Fundamentals of Python:From First Programs Through Data Structures

Chapter 18

Hierarchical Collections: Trees

objectives
Objectives

After completing this chapter, you will be able to:

  • Describe the difference between trees and other types of collections using the relevant terminology
  • Recognize applications for which general trees and binary trees are appropriate
  • Describe the behavior and use of specialized trees, such as heaps, BSTs, and expression trees
  • Analyze the performance of operations on binary search trees and heaps
  • Develop recursive algorithms to process trees

Fundamentals of Python: From First Programs Through Data Structures

an overview of trees
An Overview of Trees
  • In a tree, the ideas of predecessor and successor are replaced with those of parent and child
  • Trees have two main characteristics:
    • Each item can have multiple children
    • All items, except a privileged item called the root, have exactly one parent

Fundamentals of Python: From First Programs Through Data Structures

tree terminology
Tree Terminology

Fundamentals of Python: From First Programs Through Data Structures

slide5

Tree Terminology (continued)

Fundamentals of Python: From First Programs Through Data Structures

slide6

Tree Terminology (continued)

Note: The height of a tree containing one node is 0

By convention, the height of an empty tree is –1

Fundamentals of Python: From First Programs Through Data Structures

general trees and binary trees
General Trees and Binary Trees
  • In a binary tree, each node has at most two children:
    • The left child and the right child

Fundamentals of Python: From First Programs Through Data Structures

recursive definitions of trees
Recursive Definitions of Trees
  • A general tree is either empty or consists of a finite set of nodes T
    • Node r is called the root
    • The set T – {r} is partitioned into disjoint subsets, each of which is a general tree
  • A binary tree is either empty or consists of a root plus a left subtree and a right subtree, each of which are binary trees

Fundamentals of Python: From First Programs Through Data Structures

why use a tree
Why Use a Tree?
  • A parse tree describes the syntactic structure of a particular sentence in terms of its component parts

Fundamentals of Python: From First Programs Through Data Structures

why use a tree continued
Why Use a Tree? (continued)
  • File system structures are also tree-like

Fundamentals of Python: From First Programs Through Data Structures

why use a tree continued1
Why Use a Tree? (continued)
  • Sorted collections can also be represented as tree-like structures
    • Called a binary search tree, or BST for short
      • Can support logarithmic searches and insertions

Fundamentals of Python: From First Programs Through Data Structures

the shape of binary trees

A full binary tree contains the maximum number of nodes for a given

height H

N nodes

Height: N – 1

The Shape of Binary Trees
  • The shape of a binary tree can be described more formally by specifying the relationship between its height and the number of nodes contained in it

Fundamentals of Python: From First Programs Through Data Structures

the shape of binary trees continued
The Shape of Binary Trees (continued)
  • The number of nodes, N, contained in a full binary tree of height H is 2H+ 1 – 1
  • The height, H, of a full binary tree with N nodes is log2(N + 1) – 1
  • The maximum amount of work that it takes to access a given node in a full binary tree is O(log N)

Fundamentals of Python: From First Programs Through Data Structures

the shape of binary trees continued1
The Shape of Binary Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

three common applications of binary trees
Three Common Applications of Binary Trees
  • In this section, we introduce three special uses of binary trees that impose an ordering on their data:
    • Heaps
    • Binary search trees
    • Expression trees

Fundamentals of Python: From First Programs Through Data Structures

heaps
Heaps
  • In a min-heap each node is ≤ to both of its children
  • A max-heap places larger nodes nearer to the root
  • Heap property: Constraint on the order of nodes
  • Heap sort builds a heap from data and repeatedly removes the root item and adds it to the end of a list
  • Heaps are also used to implement priority queues

Fundamentals of Python: From First Programs Through Data Structures

binary search trees
Binary Search Trees
  • A BST imposes a sorted ordering on its nodes
    • Nodes in left subtree of a node are < node
    • Nodes in right subtree of a node are > node
  • When shape approaches that of a perfectly balanced binary tree, searches and insertions are O(log n) in the worst case
  • Not all BSTs are perfectly balanced
    • In worst case, they become linear and support linear searches

Fundamentals of Python: From First Programs Through Data Structures

binary search trees continued
Binary Search Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

binary search trees continued1
Binary Search Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

expression trees
Expression Trees
  • Another way to process expressions is to build a parse tree during parsing
    • Expression tree
  • An expression tree is never empty
  • An interior node represents a compound expression, consisting of an operator and its operands
  • Each leaf node represents a numeric operand
  • Operands of higher precedence usually appear near bottom of tree, unless overridden in source expression by parentheses

Fundamentals of Python: From First Programs Through Data Structures

expression trees continued
Expression Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

binary tree traversals
Binary Tree Traversals
  • Four standard types of traversals for binary trees:
    • Preorder traversal: Visits root node, and then traverses left subtree and right subtree in similar way
    • Inorder traversal: Traverses left subtree, visits root node, and traverses right subtree
      • Appropriate for visiting items in a BST in sorted order
    • Postorder traversal: Traverses left subtree, traverses right subtree, and visits root node
    • Level order traversal: Beginning with level 0, visits the nodes at each level in left-to-right order

Fundamentals of Python: From First Programs Through Data Structures

slide23

Binary Tree Traversals (continued)

Fundamentals of Python: From First Programs Through Data Structures

slide24

Binary Tree Traversals (continued)

Fundamentals of Python: From First Programs Through Data Structures

slide25

Binary Tree Traversals (continued)

Fundamentals of Python: From First Programs Through Data Structures

slide26

Binary Tree Traversals (continued)

Fundamentals of Python: From First Programs Through Data Structures

a binary tree adt
A Binary Tree ADT
  • Provides many common operations required for building more specialized types of trees
  • Should support basic operations for creating trees, determining if a tree is empty, and traversing a tree
  • Remaining operations focus on accessing, replacing, or removing the component parts of a nonempty binary tree—its root, left subtree, and right subtree

Fundamentals of Python: From First Programs Through Data Structures

the interface for a binary tree adt
The Interface for a Binary Tree ADT

Fundamentals of Python: From First Programs Through Data Structures

slide29

The Interface for a Binary Tree ADT (continued)

Fundamentals of Python: From First Programs Through Data Structures

processing a binary tree
Processing a Binary Tree
  • Many algorithms for processing binary trees follow the trees’ recursive structure
  • Programmers are occasionally interested in the frontier, or set of leaf nodes, of a tree
    • Example: Frontier of parse tree for English sentence shown earlier contains the words in the sentence

Fundamentals of Python: From First Programs Through Data Structures

processing a binary tree continued
Processing a Binary Tree (continued)
  • frontierexpects a binary tree and returns a list
    • Two base cases:
      • Tree is empty  return an empty list
      • Tree is a leaf node  return a list containing root item

Fundamentals of Python: From First Programs Through Data Structures

implementing a binary tree
Implementing a Binary Tree

Fundamentals of Python: From First Programs Through Data Structures

slide33

Implementing a Binary Tree (continued)

Fundamentals of Python: From First Programs Through Data Structures

slide34

Implementing a Binary Tree (continued)

Fundamentals of Python: From First Programs Through Data Structures

the string representation of a tree
The String Representation of a Tree
  • __str__can be implemented with any of the traversals

Fundamentals of Python: From First Programs Through Data Structures

developing a binary search tree
Developing a Binary Search Tree
  • A BST imposes a special ordering on the nodes in a binary tree, so as to support logarithmic searches and insertions
  • In this section, we use the binary tree ADT to develop a binary search tree, and assess its performance

Fundamentals of Python: From First Programs Through Data Structures

the binary search tree interface
The Binary Search Tree Interface
  • The interface for a BST should include a constructor and basic methods to test a tree for emptiness, determine the number of items, add an item, remove an item, and search for an item
  • Another useful method is __iter__, which allows users to traverse the items in BST with a forloop

Fundamentals of Python: From First Programs Through Data Structures

data structures for the implementation of bst

Data Structures for the Implementation of BST

Fundamentals of Python: From First Programs Through Data Structures

searching a binary search tree
Searching a Binary Search Tree
  • findreturns the first matching item if the target item is in the tree; otherwise, it returns None
    • We can use a recursive strategy

Fundamentals of Python: From First Programs Through Data Structures

inserting an item into a binary search tree
Inserting an Item into a Binary Search Tree
  • addinserts an item in its proper place in the BST
  • Item’s proper place will be in one of three positions:
    • The root node, if the tree is already empty
    • A node in the current node’s left subtree, if new item is less than item in current node
    • A node in the current node’s right subtree, if new item is greater than or equal to item in current node
  • For options 2 and 3, adduses a recursive helper function named addHelper
  • In all cases, an item is added as a leaf node

Fundamentals of Python: From First Programs Through Data Structures

removing an item from a binary search tree
Removing an Item from a Binary Search Tree
  • Save a reference to root node
  • Locate node to be removed, its parent, and its parent’s reference to this node
  • If item is not in tree, return None
  • Otherwise, if node has a left and right child, replace node’s value with largest value in left subtree and delete that value’s node from left subtree
    • Otherwise, set parent’s reference to node to node’s only child
  • Reset root node to saved reference
  • Decrement size and return item

Fundamentals of Python: From First Programs Through Data Structures

removing an item from a binary search tree continued
Removing an Item from a Binary Search Tree (continued)
  • Fourth step is fairly complex: Can be factored out into a helper function, which takes node to be deleted as a parameter (node containing item to be removed is referred to as the top node):
    • Search top node’s left subtree for node containing the largest item (rightmost node of the subtree)
    • Replace top node’s value with the item
    • If top node’s left child contained the largest item, set top node’s left child to its left child’s left child
    • Otherwise, set parent node’s right child to that right child’s left child

Fundamentals of Python: From First Programs Through Data Structures

complexity analysis of binary search trees
Complexity Analysis of Binary Search Trees
  • BSTs are set up with intent of replicating O(log n) behavior for the binary search of a sorted list
  • A BST can also provide fast insertions
  • Optimal behavior depends on height of tree
    • A perfectly balanced tree supports logarithmic searches
    • Worst case (items are inserted in sorted order): tree’s height is linear, as is its search behavior
  • Insertions in random order result in a tree with close-to-optimal search behavior

Fundamentals of Python: From First Programs Through Data Structures

case study parsing and expression trees
Case Study: Parsing and Expression Trees
  • Request:
    • Write a program that uses an expression tree to evaluate expressions or convert them to alternative forms
  • Analysis:
    • Like the parser developed in Chapter 17, current program parses an input expression and prints syntax error messages if errors occur
    • If expression is syntactically correct, program prints its value and its prefix, infix, and postfix representations

Fundamentals of Python: From First Programs Through Data Structures

slide45

Case Study: Parsing and Expression Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

slide46

Case Study: Parsing and Expression Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

case study parsing and expression trees continued
Case Study: Parsing and Expression Trees (continued)
  • Design and Implementation of the Node Classes:

Fundamentals of Python: From First Programs Through Data Structures

slide48

Case Study: Parsing and Expression Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

slide49

Case Study: Parsing and Expression Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

case study parsing and expression trees continued1
Case Study: Parsing and Expression Trees (continued)
  • Design and Implementation of the Parser Class:
    • Easiest to build an expression tree with a parser that uses a recursive descent strategy
      • Borrow parser from Chapter 17 and modify it
    • parseshould now return an expression tree to its caller, which uses that tree to obtain information about the expression
    • factorprocesses either a number or an expression nested in parentheses
      • Calls expressionto parse nested expressions

Fundamentals of Python: From First Programs Through Data Structures

slide51

Case Study: Parsing and Expression Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

slide52

Case Study: Parsing and Expression Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

an array implementation of binary trees
An Array Implementation of Binary Trees
  • An array-based implementation of a binary tree is difficult to define and practical only in some cases
  • For complete binary trees, there is an elegant and efficient array-based representation
    • Elements are stored by level
  • The array representation of a binary tree is pretty rare and is used mainly to implement a heap

Fundamentals of Python: From First Programs Through Data Structures

slide54

An Array Implementation of Binary Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

slide55

An Array Implementation of Binary Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

slide56

An Array Implementation of Binary Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

implementing heaps
Implementing Heaps

Fundamentals of Python: From First Programs Through Data Structures

implementing heaps continued
Implementing Heaps (continued)
  • At most, log2n comparisons must be made to walk up the tree from the bottom, so add is O(log n)
  • Method may trigger a doubling in the array size
    • O(n), but amortized over all additions, it is O(1)

Fundamentals of Python: From First Programs Through Data Structures

using a heap to implement a priority queue
Using a Heap to Implement a Priority Queue
  • In Ch15, we implemented a priority queue with a sorted linked list; alternatively, we can use a heap

Fundamentals of Python: From First Programs Through Data Structures

summary
Summary
  • Trees are hierarchical collections
    • The topmost node in a tree is called its root
    • In a general tree, each node below the root has at most one parent node, and zero child nodes
    • Nodes without children are called leaves
    • Nodes that have children are called interior nodes
    • The root of a tree is at level 0
  • In a binary tree, nodes have at most two children
    • A complete binary tree fills each level of nodes before moving to next level; a full binary tree includes all the possible nodes at each level

Fundamentals of Python: From First Programs Through Data Structures

summary continued
Summary (continued)
  • Four standard types of tree traversals: Preorder, inorder, postorder, and level order
  • Expression tree: Type of binary tree in which the interior nodes contain operators and the successor nodes contain their operands
  • Binary search tree: Nonempty left subtree has data < datum in its parent node and a nonempty right subtree has data > datum in its parent node
    • Logarithmic searches/insertions if close to complete
  • Heap: Binary tree in which smaller data items are located near root

Fundamentals of Python: From First Programs Through Data Structures

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