Data Structures and Algorithms
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Questions and Answers

What does the function inOrder(v) do?

  • It recursively traverses the right subtree, visits the current node, then recursively traverses the left subtree.
  • It visits the current node, then recursively traverses the left subtree, then recursively traverses the right subtree.
  • It recursively traverses the left subtree, then the right subtree, then visits the current node.
  • It recursively traverses the left subtree, visits the current node, then recursively traverses the right subtree. (correct)

    • What is the value of x(v) in the context of an inorder traversal?

  • The inorder rank of the node `v`. (correct)
  • The depth of the node `v`.
  • The number of children of the node `v`.
  • The height of the node `v`.

    • What is a common application of inorder traversal?

  • Finding the minimum value in a binary search tree.
  • Calculating the sum of all nodes in a tree.
  • Drawing a binary tree. (correct)
  • Sorting a list of elements.

    • In a proper binary tree, how many children does each internal node have?

    <p>Exactly two.</p> Signup and view all the answers

    What kind of nodes represent operators in an arithmetic expression tree?

    <p>Internal nodes.</p> Signup and view all the answers

    What is the purpose of a decision tree?

    <p>To make decisions based on a series of yes/no questions.</p> Signup and view all the answers

    What does the notation 'e' represent in the context of properties of proper binary trees?

    <p>Number of external nodes</p> Signup and view all the answers

    Which of the following expressions accurately represents the relationship between the number of nodes ('n') and the number of external nodes ('e') in a proper binary tree?

    <p>n = 2e - 1</p> Signup and view all the answers

    What does the notation 'h' represent in the context of properties of proper binary trees?

    <p>Height of the tree</p> Signup and view all the answers

    What is the relationship between the height 'h' and the number of internal nodes 'i' in a proper binary tree?

    <p>h &lt;= i</p> Signup and view all the answers

    Which of the following expressions correctly defines the relationship between the height 'h' and the number of external nodes 'e' in a proper binary tree?

    <p>e &lt;= 2h</p> Signup and view all the answers

    What is the key characteristic of a proper binary tree?

    <p>Each node has either 0 or 2 children</p> Signup and view all the answers

    Study Notes

    Data Structures and Algorithms

    • Familiarity with order of running time, Big-Oh function, and amortized analysis is assumed
    • Storing elements in a linear fashion is possible with vectors and lists
    • Position, containers, and iterators are related concepts

    Trees

    • A tree data structure is a hierarchical structure used to represent and organize data in a way that is easy to navigate and search
    • A tree consists of nodes connected by edges, with a topmost node called the root and nodes below it called child nodes
    • Each node can have multiple child nodes, forming a recursive structure

    Types of Trees

    • Binary tree: a node can have a maximum of two child nodes
    • Balanced tree: the height of the left sub-tree and the right sub-tree is equal or differs at most by 1
    • Binary search tree: used for searching and sorting algorithms

    Applications of Tree Data Structures

    • Spanning trees: used in routers to direct packets to their destination
    • Binary Search Tree: helps maintain a sorted stream of data
    • Storing hierarchical data: tree data structures are used to store data in a hierarchical order
    • Syntax tree: represents the structure of a program's source code, used in compilers
    • Heap: a tree data structure used to implement priority queues
    • Artificial intelligence: decision trees and other tree-based models are used to make predictions and classify data
    • Database: some databases use trees to organize data for efficient searching and sorting
    • Network: routing algorithms for networks use trees to find the best path for data to travel

    Tree Traversal Algorithms

    • Preorder traversal: visits a node before its descendants
    • Postorder traversal: visits a node after its descendants
    • Inorder traversal: visits a node after its left subtree and before its right subtree

    Traversal Computations

    • Depth of a node: the distance from the root to the node
    • Height of a tree: the maximum depth of its leaves

    Tree Traversal Example

    • "du" command: prints the aggregate file sizes from the current directory

    Tree Traversal Complexity

    • Preorder traversal: O(n)
    • Postorder traversal: O(n)
    • Inorder traversal: O(n)

    Implementation of Tree Traversal

    Binary Trees

    • A binary tree is a tree with each internal node having at most two children
    • Applications include arithmetic expressions and decision processes

    Trees in Data Structures

    • A tree is a hierarchical data structure used to represent and organize data in a way that is easy to navigate and search.
    • A tree consists of nodes connected by edges, with a hierarchical relationship between nodes.
    • The topmost node is called the root, and nodes below it are called child nodes.

    Types of Nodes

    • Root node: the topmost node in a tree.
    • Internal node: a node with at least one child node.
    • External node (leaf): a node without children.

    Tree Terminology

    • Ancestors of a node: parent, grandparent, grand-grandparent, etc.
    • Descendants of a node: child, grandchild, grand-grandchild, etc.
    • Depth of a node: the number of ancestors.
    • Height of a tree: the maximum depth of any node.

    Recursive Definition of a Tree

    • A tree consists of a root, and zero or more subtrees T1, T2, … , Tk, with an edge from the root to the root of each subtree.

    Preorder and Inorder Traversal

    • Preorder traversal: a node is visited before its descendants.
    • Inorder traversal: a node is visited after its left subtree and before its right subtree.

    Binary Trees

    • A binary tree is a tree with each internal node having at most two children (exactly two for proper binary trees).
    • Applications: arithmetic expressions, decision processes, searching.

    Properties of Proper Binary Trees

    • Notation: n (number of nodes), e (number of external nodes), i (number of internal nodes), h (height).
    • Properties: e = i + 1, n = 2e - 1, h ≤ i, h ≤ (n - 1)/2, e ≤ 2h, h ≥ log2 e, h ≥ log2 (n + 1) - 1.

    BinaryTree ADT

    • Extends the Tree ADT, inheriting all its methods.
    • Additional methods: position p.left(), position p.right().
    • Proper binary tree: each node has either 0 or 2 children.

    Evaluating Arithmetic Expressions

    • Specialization of a postorder traversal, returning the value of a subtree.
    • Recursive method combining the values of subtrees when visiting an internal node.

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    Quiz Team

    Description

    This quiz covers data structures and algorithms, including vectors, lists, and trees. Familiarity with Big-Oh function and amortized analysis is assumed.

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