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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

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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

• 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

• 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

Fundamentals of Python: From First Programs Through Data Structures

Tree Terminology (continued)

Fundamentals of Python: From First Programs Through Data Structures

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

• 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

• 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?

• 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)

• File system structures are also tree-like

Fundamentals of Python: From First Programs Through Data Structures

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

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 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 (continued)

Fundamentals of Python: From First Programs Through Data Structures

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

• 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

• 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)

Fundamentals of Python: From First Programs Through Data Structures

Binary Search Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

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)

Fundamentals of Python: From First Programs Through Data Structures

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

Binary Tree Traversals (continued)

Fundamentals of Python: From First Programs Through Data Structures

Binary Tree Traversals (continued)

Fundamentals of Python: From First Programs Through Data Structures

Binary Tree Traversals (continued)

Fundamentals of Python: From First Programs Through Data Structures

Binary Tree Traversals (continued)

Fundamentals of Python: From First Programs Through Data Structures

• 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

Fundamentals of Python: From First Programs Through Data Structures

The Interface for a Binary Tree ADT (continued)

Fundamentals of Python: From First Programs Through Data Structures

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)

• 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

Fundamentals of Python: From First Programs Through Data Structures

Implementing a Binary Tree (continued)

Fundamentals of Python: From First Programs Through Data Structures

Implementing a Binary Tree (continued)

Fundamentals of Python: From First Programs Through Data Structures

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

• 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 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

Fundamentals of Python: From First Programs Through Data Structures

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

• 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

• 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)

• 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

• 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

• 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

Case Study: Parsing and Expression Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

Case Study: Parsing and Expression Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

Case Study: Parsing and Expression Trees (continued)

• Design and Implementation of the Node Classes:

Fundamentals of Python: From First Programs Through Data Structures

Case Study: Parsing and Expression Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

Case Study: Parsing and Expression Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

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

Case Study: Parsing and Expression Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

Case Study: Parsing and Expression Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

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

An Array Implementation of Binary Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

An Array Implementation of Binary Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

An Array Implementation of Binary Trees (continued)

Fundamentals of Python: From First Programs Through Data Structures

Implementing Heaps

Fundamentals of Python: From First Programs Through Data Structures

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

• 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

• 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)

• 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