09 binary-trees

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This document discusses binary trees, including their basic structure and components. A binary tree is composed of nodes, with each node containing a value and references to a left and right child node. It describes traversing binary trees in preorder, inorder, and postorder sequences by recursively visiting each node. Various properties are also covered, such as a tree's size, depth, balance, and copying an entire binary tree using postorder traversal.

Parts of a binary tree
 A binary tree is composed of zero or more nodes
 Each node contains:
 A value (some sort of data item)
 A reference or pointer to a left child (may be null), and
 A reference or pointer to a right child (may be null)
 A binary tree may be empty (contain no nodes)
 If not empty, a binary tree has a root node
 Every node in the binary tree is reachable from the root
node by a unique path
 A node with neither a left child nor a right child is
called a leaf
 In some binary trees, only the leaves contain a value
2

Picture of a binary tree

a

b c

d e f

g h i j k

l
3

Size and depth
 The size of a binary tree is the
a number of nodes in it
 This tree has size 12
b c  The depth of a node is its
distance from the root
d e f

a is at depth zero

e is at depth 2
g h i j k
 The depth of a binary tree is
l the depth of its deepest node
 This tree has depth 4

4

Balance
a a
b c b
c e
d e f g
d f
h i j
g h
A balanced binary tree
i j
An unbalanced binary tree
 A binary tree is balanced if every level above the lowest is “full”
(contains 2n nodes)
 In most applications, a reasonably balanced binary tree is
desirable
5

Binary search in an array
 Look at array location (lo + hi)/2

Searching for 5:
(0+6)/2 = 3
Using a binary
hi = 2;
(0 + 2)/2 = 1 lo = 2; search tree
(2+2)/2=2
7

3 13
0 1 2 3 4 5 6
2 3 5 7 11 13 17 2 5 11 17

6

Tree traversals
 A binary tree is defined recursively: it consists of a root, a
left subtree, and a right subtree
 To traverse (or walk) the binary tree is to visit each node in
the binary tree exactly once
 Tree traversals are naturally recursive
 Since a binary tree has three “parts,” there are six possible
ways to traverse the binary tree:
 root, right, left
 root, left, right
 right, root, left
 left, root, right
 right, left, root
 left, right, root

7

Preorder traversal
 In preorder, the root is visited first
 Here’s a preorder traversal to print out all the
elements in the binary tree:

public void preorderPrint(BinaryTree bt) {
if (bt == null) return;
System.out.println(bt.value);
preorderPrint(bt.leftChild);
preorderPrint(bt.rightChild);
}

8

Inorder traversal
 In inorder, the root is visited in the middle
 Here’s an inorder traversal to print out all the
elements in the binary tree:

public void inorderPrint(BinaryTree bt) {
if (bt == null) return;
inorderPrint(bt.leftChild);
System.out.println(bt.value);
inorderPrint(bt.rightChild);
}

9

Postorder traversal
 In postorder, the root is visited last
 Here’s a postorder traversal to print out all the
elements in the binary tree:

public void postorderPrint(BinaryTree bt) {
if (bt == null) return;
postorderPrint(bt.leftChild);
postorderPrint(bt.rightChild);
System.out.println(bt.value);
}

10

Tree traversals using “flags”
 The order in which the nodes are visited during a tree
traversal can be easily determined by imagining there is a
“flag” attached to each node, as follows:

preorder inorder postorder

 To traverse the tree, collect the flags:
A A A

B C B C B C

D E F G D E F G D E F G

ABDECFG DBEAFCG DEBFGCA
11

Copying a binary tree
 In postorder, the root is visited last
 Here’s a postorder traversal to make a complete
copy of a given binary tree:

public BinaryTree copyTree(BinaryTree bt) {
if (bt == null) return null;
BinaryTree left = copyTree(bt.leftChild);
BinaryTree right = copyTree(bt.rightChild);
return new BinaryTree(bt.value, left, right);
}

12

Other traversals
 The other traversals are the reverse of these three
standard ones
 That is, the right subtree is traversed before the left subtree
is traversed
 Reverse preorder: root, right subtree, left subtree
 Reverse inorder: right subtree, root, left subtree
 Reverse postorder: right subtree, left subtree, root

13

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The document discusses strings in C and common string functions. It defines a string as an array of characters terminated by a null character. It describes two ways to use strings - with a character array or string pointer. It then explains functions such as strcpy(), strcat(), strcmp() that copy, append, or compare strings. Other functions like memcpy(), memcmp() operate on a specified number of characters rather than null-terminated strings.

Statistical quality__control_2Tech_MX

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The document discusses implementation of stacks. It describes stacks as linear data structures that follow LIFO principles. Key stack operations like push and pop are outlined. Stacks are often implemented using arrays or linked lists. Examples of stack applications include recursion handling, expression evaluation, parenthesis checking, and backtracking problems. Conversion from infix to postfix notation using a stack is also demonstrated.

Stack Data Structure & It's Application Tech_MX

Stack is a last-in, first-out (LIFO) data structure where elements are inserted and removed from the top. Pushing adds an element to the top of the stack, while popping removes the top element. A stack overflow occurs when pushing to a full stack, while a stack underflow happens when popping an empty stack. Stack applications include system startup/shutdown processes, function calling where the last function called is the first to return, and argument passing in C where arguments are pushed right-to-left and popped left-to-right.

Spanning trees & applicationsTech_MX

This document discusses minimum spanning trees. It defines a minimum spanning tree as a spanning tree of a connected, undirected graph that has a minimum total cost among all spanning trees of that graph. The document provides properties of minimum spanning trees, including that they are acyclic, connect all vertices, and have n-1 edges for a graph with n vertices. Applications of minimum spanning trees mentioned include communication networks, power grids, and laying telephone wires to minimize total length.

Real time Operating SystemTech_MX

This document discusses real-time operating systems (RTOS). It defines RTOS as operating systems that are able to respond to inputs immediately within a specified time delay. It compares RTOS to general operating systems and discusses the types, characteristics, functions, and applications of RTOS. Examples of RTOS like VxWorks are provided. The key functions of an RTOS include task management, scheduling, resource allocation, and interrupt handling. RTOS are widely used in applications that require deterministic responses like avionics, medical devices, industrial automation, and more.

ParsingTech_MX

The document discusses parsing in compilers. It defines parsing as the second stage after lexical analysis, where a parser checks if the stream of tokens from the lexical analyzer is grammatically correct by generating a parse tree. The fundamental theory behind parsing is context-free grammar, which is used to define languages and check parsing. The document then discusses context-free grammars, parse trees, ambiguity, and provides examples of grammars for Boolean expressions to illustrate parsing concepts.

Mouse interrupts (Assembly Language & C)Tech_MX

This document discusses interrupts and the mouse on the 8086 CPU. It describes how interrupts work on the 8086, including hardware, software, and interrupt service routines. It explains the fetch-execute cycle of the CPU and the interrupt sequence. It then covers mouse features like pixels and the mouse pointer. It discusses the interrupt vector table format and how mouse functions like initializing, displaying, hiding the cursor, and getting button/position information are accessed through interrupt 33h.

Motherboard of a pcTech_MX

The motherboard is the central circuit board in a computer that connects the various components like the CPU, memory, storage, and peripherals. It provides the electrical pathways allowing these components to communicate with each other. Over time, more functions have been integrated onto motherboards, from basic components in early computers to modern boards that support complex graphics, audio, and networking. The motherboard contains the BIOS, chipset that directs data flow, and expansion slots to connect additional components. Common motherboard failures include catastrophic failures during early use, intermittent component failures, and difficult to diagnose issues causing crashes or reboots.

MultiMedia dbmsTech_MX

A multimedia database management system (MMDBMS) stores and manages multimedia data like text, images, audio, and video. Multimedia data can be stored in three parts - raw data, registering data, and descriptive data. Registering data provides interpretation information, descriptive data describes content and structure, and raw data is the unformatted information content like pixels. An MMDBMS allows input, output, modification, deletion, comparison, and evaluation operations on different multimedia data types. MMDBMS design can be based on object-oriented or entity-relationship database models.

Merging files (Data Structure)Tech_MX

This document discusses different techniques for merging files in revision control systems. It begins by introducing the concept of merging as reconciling multiple changes made to files. It then discusses external sorting techniques that can handle large amounts of data. The main merging techniques covered are two-way merging, three-way merging, and k-way merging. Two-way merging considers differences between two files alone, while three-way merging also looks at the original parent file. Three-way merging is generally more reliable with less need for user intervention. K-way merging uses a tournament sort algorithm to merge an arbitrary number of files.

Memory dbmsTech_MX

This document discusses memory-based database management systems (MDBMS). Key points include: - An MDBMS stores the database in main memory rather than disk storage for faster access speed. However, data is transient and could be lost if power is lost. - MDBMS are well-suited for applications with frequent data reads, shared databases with many users, or where performance is critical. They are less suitable when data persistence is required. - Sybase implemented an MDBMS that uses memory as a virtual disk volume, retaining the SQL interface. Transactions are stored in a transfer table then committed to the original disk-based database.

Linear regressionTech_MX

This document provides an example of simple linear regression with one independent variable. It explains that linear regression finds the line of best fit by estimating values for the slope (b1) and y-intercept (b0) that minimize the sum of the squared errors between the observed data points and the regression line. It provides the formulas for calculating the least squares estimates of b1 and b0. The document includes a table of temperature and sales data and a corresponding scatter plot as an example of simple linear regression analysis.

Keyboard interruptTech_MX

The document discusses various keyboard and video interrupts used in 8086 processors. It explains that interrupts allow efficient processing of I/O without wasting CPU time waiting. INT 21h is used for keyboard and file operations depending on the value in register AH. INT 16h provides keyboard services like reading keys. INT 10h controls video modes and can set pixels, write text, and get display information. Example programs demonstrate using interrupts to get input, output text, and retrieve the date.

Introduction to loadersTech_MX

The document discusses loaders, which are system software programs that perform the loading function of placing a program into memory for execution. There are several types of loaders: compile-and-go loaders directly place assembled code into memory; absolute loaders place code at specified addresses; relocating loaders allow code to be loaded at different addresses and combine programs. Relocating loaders output object code, symbol tables, and relocation information to perform allocation, relocation, linking, and loading separately from assembly. Direct-linking loaders provide more flexibility by allowing multiple program and data segments with intersegment references.

Interpersonal communicationTech_MX

Interpersonal communication involves the exchange of verbal and non-verbal messages between people. It includes skills like listening, problem solving, decision making, and stress management. Effective interpersonal communication is important for maintaining good relationships both personally and professionally. Some key aspects of interpersonal communication are listening skills, verbal and non-verbal communication skills, and assertiveness. Listening requires focusing not just on the words but also how the message is delivered. Both verbal and non-verbal elements contribute to effective communication. Being assertive means standing up for yourself while also respecting others.

Interesting applications of graph theoryTech_MX

The document discusses graphs and their applications. It defines key graph terms like vertices, edges, directed/undirected graphs, paths, cycles, etc. It then describes algorithms for finding minimum spanning trees, Eulerian cycles, Hamiltonian paths, and approximations for the traveling salesman problem. Examples are provided to illustrate minimum spanning tree and Christofide's algorithms for TSP.

Virtual base classTech_MX

Theory of estimationTech_MX

Templates in C++Tech_MX

String & its applicationTech_MX

Statistical quality__control_2Tech_MX

Stack data structureTech_MX

Stack Data Structure & It's Application Tech_MX

Spanning trees & applicationsTech_MX

Real time Operating SystemTech_MX

ParsingTech_MX

Mouse interrupts (Assembly Language & C)Tech_MX

Motherboard of a pcTech_MX

MultiMedia dbmsTech_MX

Merging files (Data Structure)Tech_MX

Memory dbmsTech_MX

Linear regressionTech_MX

Keyboard interruptTech_MX

Introduction to loadersTech_MX

Interpersonal communicationTech_MX

Interesting applications of graph theoryTech_MX

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09 binary-trees

1. Binary Trees

2. Parts of a binary tree  A binary tree is composed of zero or more nodes  Each node contains:  A value (some sort of data item)  A reference or pointer to a left child (may be null), and  A reference or pointer to a right child (may be null)  A binary tree may be empty (contain no nodes)  If not empty, a binary tree has a root node  Every node in the binary tree is reachable from the root node by a unique path  A node with neither a left child nor a right child is called a leaf  In some binary trees, only the leaves contain a value 2

3. Picture of a binary tree a b c d e f g h i j k l 3

4. Size and depth  The size of a binary tree is the a number of nodes in it  This tree has size 12 b c  The depth of a node is its distance from the root d e f  a is at depth zero  e is at depth 2 g h i j k  The depth of a binary tree is l the depth of its deepest node  This tree has depth 4 4

5. Balance a a b c b c e d e f g d f h i j g h A balanced binary tree i j An unbalanced binary tree  A binary tree is balanced if every level above the lowest is “full” (contains 2n nodes)  In most applications, a reasonably balanced binary tree is desirable 5

6. Binary search in an array  Look at array location (lo + hi)/2 Searching for 5: (0+6)/2 = 3 Using a binary hi = 2; (0 + 2)/2 = 1 lo = 2; search tree (2+2)/2=2 7 3 13 0 1 2 3 4 5 6 2 3 5 7 11 13 17 2 5 11 17 6

7. Tree traversals  A binary tree is defined recursively: it consists of a root, a left subtree, and a right subtree  To traverse (or walk) the binary tree is to visit each node in the binary tree exactly once  Tree traversals are naturally recursive  Since a binary tree has three “parts,” there are six possible ways to traverse the binary tree:  root, right, left  root, left, right  right, root, left  left, root, right  right, left, root  left, right, root 7

8. Preorder traversal  In preorder, the root is visited first  Here’s a preorder traversal to print out all the elements in the binary tree: public void preorderPrint(BinaryTree bt) { if (bt == null) return; System.out.println(bt.value); preorderPrint(bt.leftChild); preorderPrint(bt.rightChild); } 8

9. Inorder traversal  In inorder, the root is visited in the middle  Here’s an inorder traversal to print out all the elements in the binary tree: public void inorderPrint(BinaryTree bt) { if (bt == null) return; inorderPrint(bt.leftChild); System.out.println(bt.value); inorderPrint(bt.rightChild); } 9

10. Postorder traversal  In postorder, the root is visited last  Here’s a postorder traversal to print out all the elements in the binary tree: public void postorderPrint(BinaryTree bt) { if (bt == null) return; postorderPrint(bt.leftChild); postorderPrint(bt.rightChild); System.out.println(bt.value); } 10

11. Tree traversals using “flags”  The order in which the nodes are visited during a tree traversal can be easily determined by imagining there is a “flag” attached to each node, as follows: preorder inorder postorder  To traverse the tree, collect the flags: A A A B C B C B C D E F G D E F G D E F G ABDECFG DBEAFCG DEBFGCA 11

12. Copying a binary tree  In postorder, the root is visited last  Here’s a postorder traversal to make a complete copy of a given binary tree: public BinaryTree copyTree(BinaryTree bt) { if (bt == null) return null; BinaryTree left = copyTree(bt.leftChild); BinaryTree right = copyTree(bt.rightChild); return new BinaryTree(bt.value, left, right); } 12

13. Other traversals  The other traversals are the reverse of these three standard ones  That is, the right subtree is traversed before the left subtree is traversed  Reverse preorder: root, right subtree, left subtree  Reverse inorder: right subtree, root, left subtree  Reverse postorder: right subtree, left subtree, root 13

14. The End 14

09 binary-trees

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09 binary-trees