CSC 1061: More Trees Quiz
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Questions and Answers

What is the primary advantage of using a B-tree over traditional binary search trees?

  • B-trees are less efficient for storing large amounts of data.
  • B-trees decrease the height of the tree, allowing faster disk accesses. (correct)
  • B-trees require more disk accesses for data retrieval.
  • B-trees can store only single key per node.

Which property of a B-tree ensures that all leaves are at the same depth?

  • All leaves have the same depth, defined as the height of the tree. (correct)
  • Each internal node can have a maximum of MAX-1 keys.
  • All leaves must contain at least one key.
  • The root must contain at least 2 children.

What is the minimum number of child nodes that each internal node (except the root) in a B-tree can have?

  • At least MAX keys.
  • At least MAX/2 children. (correct)
  • At least 1 key.
  • At least 2 children.

How does searching for a value in a B-tree typically work?

<p>Using binary tree search followed by searching the array inside the node. (A)</p> Signup and view all the answers

Which scenario best exemplifies the use of a B-tree?

<p>Manipulating a large, undetermined amount of data in a database system. (D)</p> Signup and view all the answers

Which traversal method visits the nodes in the order of left child, parent, and then right child?

<p>In-Order (C)</p> Signup and view all the answers

What is a distinguishing property of a binary search tree compared to a regular binary tree?

<p>All nodes of the left subtree are less than or equal to the parent node. (C)</p> Signup and view all the answers

Which of the following is the correct order for a pre-order traversal?

<p>4, 2, 5, 1, 3 (D)</p> Signup and view all the answers

Which tree type is characterized by having filled levels and only the last level occupied from left to right?

<p>Complete Tree (C)</p> Signup and view all the answers

In the context of passing a function as an argument, which option represents the function signature correctly?

<p>returnType functionName(dataType, ...) (C)</p> Signup and view all the answers

Which of these traversal methods lists the nodes in a manner where the parent node is visited after its children?

<p>Post-Order (D)</p> Signup and view all the answers

Which of the following sequences corresponds to the backwards in-order traversal?

<p>1, 2, 4, 5, 3 (C)</p> Signup and view all the answers

What is the process called to create a heap data structure from a binary tree?

<p>Heapify (C)</p> Signup and view all the answers

In a Min-Heap, how does the relationship between parent and child nodes work?

<p>Each parent node is smaller than its child nodes. (C)</p> Signup and view all the answers

Which indices correctly represent the parent and child nodes in a binary tree stored as an array?

<p>Parent: array[(i - 1) / 2], Left child: array[2 * i + 1], Right child: array[2 * i + 2] (D)</p> Signup and view all the answers

What is a defining characteristic of a B-tree?

<p>Each node can contain more than one key and more than two children. (B)</p> Signup and view all the answers

Which types of trees are NOT mentioned in relation to binary trees?

<p>B-Trees and Red-Black Trees (C)</p> Signup and view all the answers

What occurs during re-heapification upward in a heap?

<p>Nodes are compared and reordered to maintain heap properties. (C)</p> Signup and view all the answers

Which statement is true for a complete binary tree?

<p>The lowest level can have gaps in its leaves. (A)</p> Signup and view all the answers

In a full binary tree, how many children does each parent node have?

<p>Either two children or zero children. (B)</p> Signup and view all the answers

What does the inOrder function do in a binary tree traversal?

<p>Visits nodes in left-root-right order. (C)</p> Signup and view all the answers

Which is the correct representation of relationships in a binary tree stored as an array with root at index 0?

<p>Current node: array[i], Parent: array[(i - 1) / 2], Children: array[2<em>i + 1] and array[2</em>i + 2] (D)</p> Signup and view all the answers

Flashcards

B-tree

A type of self-balancing search tree optimized for disk-based storage, where each node can hold multiple keys and pointers to child nodes.

Order of a B-tree

The maximum number of keys a node in a B-tree can hold. The order of a B-tree determines the minimum and maximum number of children a node can have.

Leaf node

A node in a B-tree that does not have any children. Leaf nodes store the actual data values.

Internal node

Nodes in a B-tree that are not leaf nodes. They contain keys and pointers to child nodes, acting as an index for navigating the tree.

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Same depth of all leaf nodes

The depth of all leaf nodes in a B-tree is the same. This ensures that the tree remains balanced and efficient for searching.

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In-Order Traversal

A binary tree traversal where the left subtree, the root, and then the right subtree are processed in that order. It produces a sorted output for sorted input.

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Pre-Order Traversal

A binary tree traversal where the root node, the left subtree, and then the right subtree are processed in that order.

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Post-Order Traversal

A binary tree traversal where the left subtree, the right subtree, and then the root node are processed in that order.

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Backward In-Order Traversal

A binary tree traversal where the right subtree, the root, and then the left subtree are processed in that order. It produces a reversed sorted output for sorted input.

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Full Binary Tree

A binary tree where all nodes have either zero or two children (no single children allowed).

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Complete Binary Tree

A binary tree where all levels are completely filled except possibly the last level. The nodes on the last level are filled from left to right.

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Heap Tree

A binary tree that is also a heap. This means it satisfies the heap property, where the parent node's value is always greater than or equal to (or less than or equal to) that of its children.

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Binary Tree

A tree data structure where each node has at most two children, referred to as the left and right child.

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Binary Search Tree

A binary tree where the left child of a node always contains a value smaller than the node's value, and the right child always contains a value larger than the node's value.

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Binary Trees as Arrays

A way to represent a binary tree using an array where each node's position corresponds to a specific index in the array.

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Max Heap

A binary tree where every node's value is greater than its children's values.

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Min Heap

A binary tree where every node's value is smaller than its children's values.

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Heapify

A process of organizing the elements of a binary tree to satisfy the heap property (either max or min heap).

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Study Notes

Course Information

  • Course code: CSC 1061
  • Topic: More Trees

Objectives

  • Follow and explain tree-based algorithms
  • Explain common binary tree traversals (in-order, pre-order, post-order, backwards in-order)
  • List binary search tree rules and determine if a tree satisfies them
  • Perform searches and insertions on binary search trees
  • Implement algorithms using binary tree class

Agenda - Week 13

  • Review Binary Tree
  • Review Binary Search Tree
  • Binary Tree as Array
  • Function as Arguments
  • Other Trees (Full Tree, Complete Tree, Heap, B-Trees)

Binary Tree Traversals

  • A binary tree is a data structure where each parent node has at most two children.
  • Traversal names match ordered sets:
    • In-Order: 4, 5, 2, 3, 1
    • Pre-Order: 4, 2, 5, 1, 3
    • Post-Order: 3, 1, 5, 2, 4
    • Backward In-Order: 1, 2, 4, 5, 3

Binary Search Tree

  • A data structure that efficiently maintains a sorted list.
  • Properties:
    • All left subtree nodes are less than or equal to the parent node
    • All right subtree nodes are greater than the parent node

Passing a Function as an Argument

  • Templates don't know about special processing for one data type.
  • To keep templates generic, functions can take other functions as arguments.
  • To pass a function as an argument, specify the full function signature in the prototype and header.
  • Only the function name is passed, but it must match the signature of the function being passed.

Binary Tree Traversal: Generic Process

  • process is a placeholder function, not a real function.
  • inOrder function performs in-order traversal using recursion.

Binary Trees as Arrays

  • Binary trees in arrays offer direct random access.
  • Each node has a direct index, allowing easy access to parents and children based on the current node's index.
  • The root node is at index [0]

Other Types of Trees

  • Besides binary search trees, there are:
    • Full Trees
    • Heaps
    • Complete Trees
    • B-Trees

Full Binary Tree

  • Every parent node has either two or no children.

Complete Binary Tree

  • All levels are filled except possibly the lowest one (which is filled from left to right).

Heap

  • A complete binary tree that satisfies either max-heap or min-heap properties.
  • Every level, excluding possibly the last, is filled.
  • All nodes are as far left as possible in the tree.

Max Heap Property

  • Each node is always greater than its child nodes.
  • The root node has the largest key among all other nodes.

Min Heap Property

  • Each node is always smaller than its child nodes.
  • The root node has the smallest key among all other nodes.

Heapify

  • Creating a heap data structure from a binary tree (creating a min-heap or max-heap).
  • Adding an element involves re-heapification upward.
  • Removing an element involves re-heapification downward.

B-Tree

  • A self-balancing search tree where each node can contain more than one key and has more than two children.
  • It's a generalization of a binary search tree.

Why B-Tree

  • The need for B-trees arose due to the need for faster access to physical storage media (like hard disks).
  • Secondary storage devices are slower, and B-trees can store multiple keys in a single node and have multiple children, significantly reducing the height of the tree (which improves access time).

B-Tree Properties

  • Keys in a node are stored in increasing order.
  • Each node has a boolean value x.leaf, which is true if the node is a leaf.
  • An internal node holds a maximum of MAX - 1 keys and pointers to each child.
  • Nodes except the root have a maximum of MAX children and a minimum of MAX/2 children.
  • All leaves have the same depth.
  • The root node has at least two children and contains one key.

B-Tree Idea

  • B-trees combine the advantages of dynamic memory and static arrays.
  • Searches use binary tree traversal to find the range containing the value, then a standard array-searching algorithm like a binary search.

Post-Review

  • Review glossary terms and complete matching questions.

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CSC 1061 Week 13 More Trees PDF

Description

This quiz covers key concepts from CSC 1061, focusing on tree-based algorithms, binary tree traversals, and binary search tree properties. You will be tested on your ability to identify traversal types, perform search and insertion operations, and understand various tree structures. Prepare to demonstrate your understanding of binary trees and their applications.

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