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
2
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
3
Tree Terminology
Fundamentals of Python: From First Programs Through Data Structures
4
Tree Terminology (continued)
Fundamentals of Python: From First Programs Through Data Structures
5
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
6
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
7
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
8
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
9
Why Use a Tree? (continued)
• File system structures are also tree-like
Fundamentals of Python: From First Programs Through Data Structures
10
Why Use a Tree? (continued)
• Sorted collections can also be represented as treelike 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
11
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
A full binary tree contains
the maximum number of
nodes for a given
height H
N nodes
Height: N – 1
Fundamentals of Python: From First Programs Through Data Structures
12
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
13
The Shape of Binary Trees (continued)
Fundamentals of Python: From First Programs Through Data Structures
14
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
15
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
16
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
17
Binary Search Trees (continued)
Fundamentals of Python: From First Programs Through Data Structures
18
Binary Search Trees (continued)
Fundamentals of Python: From First Programs Through Data Structures
19
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
20
Expression Trees (continued)
Fundamentals of Python: From First Programs Through Data Structures
21
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
22
Binary Tree Traversals (continued)
Fundamentals of Python: From First Programs Through Data Structures
23
Binary Tree Traversals (continued)
Fundamentals of Python: From First Programs Through Data Structures
24
Binary Tree Traversals (continued)
Fundamentals of Python: From First Programs Through Data Structures
25
Binary Tree Traversals (continued)
Fundamentals of Python: From First Programs Through Data Structures
26
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
27
The Interface for a Binary Tree ADT
Fundamentals of Python: From First Programs Through Data Structures
28
The Interface for a Binary Tree ADT
(continued)
Fundamentals of Python: From First Programs Through Data Structures
29
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
30
Processing a Binary Tree (continued)
• frontier expects 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
31
Implementing a Binary Tree
Fundamentals of Python: From First Programs Through Data Structures
32
Implementing a Binary Tree
(continued)
Fundamentals of Python: From First Programs Through Data Structures
33
Implementing a Binary Tree
(continued)
Fundamentals of Python: From First Programs Through Data Structures
34
The String Representation of a Tree
• __str__ can be implemented with any of the
traversals
Fundamentals of Python: From First Programs Through Data Structures
35
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
36
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 for loop
Fundamentals of Python: From First Programs Through Data Structures
37
Data Structures for the Implementation
of BST
…
Fundamentals of Python: From First Programs Through Data Structures
38
Searching a Binary Search Tree
• find returns 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
39
Inserting an Item into a Binary Search
Tree
• add inserts 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, add uses 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
40
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
41
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
42
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
43
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
44
Case Study: Parsing and Expression
Trees (continued)
Fundamentals of Python: From First Programs Through Data Structures
45
Case Study: Parsing and Expression
Trees (continued)
Fundamentals of Python: From First Programs Through Data Structures
46
Case Study: Parsing and Expression
Trees (continued)
• Design and Implementation of the Node Classes:
Fundamentals of Python: From First Programs Through Data Structures
47
Case Study: Parsing and Expression
Trees (continued)
Fundamentals of Python: From First Programs Through Data Structures
48
Case Study: Parsing and Expression
Trees (continued)
Fundamentals of Python: From First Programs Through Data Structures
49
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
– parse should now return an expression tree to its
caller, which uses that tree to obtain information
about the expression
– factor processes either a number or an expression
nested in parentheses
• Calls expression to parse nested expressions
Fundamentals of Python: From First Programs Through Data Structures
50
Case Study: Parsing and Expression
Trees (continued)
Fundamentals of Python: From First Programs Through Data Structures
51
Case Study: Parsing and Expression
Trees (continued)
Fundamentals of Python: From First Programs Through Data Structures
52
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
53
An Array Implementation of Binary
Trees (continued)
Fundamentals of Python: From First Programs Through Data Structures
54
An Array Implementation of Binary
Trees (continued)
Fundamentals of Python: From First Programs Through Data Structures
55
An Array Implementation of Binary
Trees (continued)
Fundamentals of Python: From First Programs Through Data Structures
56
Implementing Heaps
Fundamentals of Python: From First Programs Through Data Structures
57
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
58
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
59
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
60
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
61

Reading from course text:
Qu es
ti ons?
______________________
Devon M. Simmonds
Computer Science Department
University of North Carolina Wilmington
_____________________________________________________________
62