Binary Search Tree Basics
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Questions and Answers

What is the first step when inserting a new key in a binary search tree (BST)?

  • Insert the new node at the root node.
  • Start searching from the root until a leaf node is found. (correct)
  • Always add the new node as a right child.
  • Start searching from the rightmost node.

When removing a node with two children from a BST, what is the appropriate action according to the algorithm?

  • Link the child directly to the surrounding nodes.
  • Remove the node and shift the remaining nodes to the left.
  • Copy the contents of the inorder successor to the node being removed. (correct)
  • Delete the node and replace it with the maximum value in the left subtree.

Which case scenario describes a situation where a node to be removed has no children?

  • Transfer the children to an external queue.
  • The parent's link should be set to NULL. (correct)
  • A new child must be added.
  • The node can be replanted.

What indicates the next node in an inorder traversal of a binary search tree?

<p>The successor node. (B)</p> Signup and view all the answers

In the context of deletion in a BST, what must be ensured when removing a node with one child?

<p>The single child must be linked directly to the parent of the removed node. (C)</p> Signup and view all the answers

Which of the following statements about binary search trees (BST) is true?

<p>All left descendants of a node are smaller than the node. (D)</p> Signup and view all the answers

What does the 'TREE-MINIMUM' function return in the context of BST?

<p>The node with the minimum value in the entire tree. (D)</p> Signup and view all the answers

Which traversal method is used to find the successor in a binary search tree?

<p>Inorder traversal (A)</p> Signup and view all the answers

What is the expected time complexity for lookup, insertion, or deletion operations in a balanced binary search tree?

<p>O(log n) (A)</p> Signup and view all the answers

Which traversal method processes the nodes in the order of parent node first, then left child, and finally right child?

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

In a binary search tree, where would you insert a new node with a value less than the root's key?

<p>In the left subtree (C)</p> Signup and view all the answers

What should be done when deleting a node from a BST that has two children?

<p>Replace it with its in-order predecessor or successor (D)</p> Signup and view all the answers

What property of a binary search tree ensures that each subtree must also be a binary search tree?

<p>There are no duplicate nodes (A)</p> Signup and view all the answers

Which of the following statements about the height of a binary search tree is true?

<p>Average height is O(log n) but worst case can be O(n) (C)</p> Signup and view all the answers

During postorder traversal of a binary search tree, which node is visited last?

<p>The root node (C)</p> Signup and view all the answers

What is a key characteristic of a binary search tree that differentiates it from a regular binary tree?

<p>All nodes are organized by a specific value order (C)</p> Signup and view all the answers

If a binary search tree has a height of O(n), what is likely true about its structure?

<p>It is a linked list (C)</p> Signup and view all the answers

Which function accurately represents the process of deleting a node in a binary search tree?

<p>Replace the node correctly based on its children (C)</p> Signup and view all the answers

What is the primary purpose of the TREE-DELETE-WITHOUT-PARENT function in a BST?

<p>To delete a node while maintaining the tree structure (A)</p> Signup and view all the answers

Which sequence of nodes is traversed first in an inorder traversal of a BST?

<p>Left, Root, Right (B)</p> Signup and view all the answers

After deleting a node with two children from a BST, what is usually the next step?

<p>Replace it with the inorder predecessor or successor (C)</p> Signup and view all the answers

What is the output of a preorder traversal of the sequence 15, 18, 6, 7, 17, 3, 4, 13, 9, 20?

<p>15, 18, 7, 6, 3, 4, 13, 9, 17, 20 (B)</p> Signup and view all the answers

When traversing a BST in postorder, which node is visited last?

<p>The root node (A)</p> Signup and view all the answers

After deleting node 4 from the given BST, what key will replace it if node 4 has a right child?

<p>The leftmost node in the right subtree (A)</p> Signup and view all the answers

Which traversal method first processes the root node?

<p>Preorder traversal (B)</p> Signup and view all the answers

What is a key characteristic of a binary search tree (BST)?

<p>All left descendants are less than the node and all right descendants are greater (B)</p> Signup and view all the answers

What will be the inorder traversal of a BST after sequentially deleting nodes 4, 7, and 15 from it?

<p>A, B, C, D, E, G, H, I (D)</p> Signup and view all the answers

What represents the minimum key during deletion in a BST?

<p>The leftmost leaf node (D)</p> Signup and view all the answers

If a BST node has no children, what occurs during its deletion?

<p>It is simply removed with no further changes (C)</p> Signup and view all the answers

When finding the inorder predecessor of a node in a BST, what must be looked at?

<p>The largest node in its left subtree (C)</p> Signup and view all the answers

In a BST, which traversal method yields nodes in sorted order?

<p>Inorder (A)</p> Signup and view all the answers

In the procedural algorithm for deleting a node, which condition is checked first?

<p>If the current node is NULL (C)</p> Signup and view all the answers

Which of the following is a characteristic of a postorder traversal?

<p>It processes child nodes before the parent (C)</p> Signup and view all the answers

Flashcards

Maximum Node

The node with the largest key value in a Binary Search Tree (BST).

Minimum Node

The node with the smallest key value in a BST.

Successor (BST)

The node immediately after a given node in an inorder traversal of a BST.

Predecessor (BST)

The node immediately before a given node in an inorder traversal of a BST.

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Tree Insertion

Adding a new node to a BST while maintaining the BST property.

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Tree Deletion (0 children)

Removing a node with no children from a BST.

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Tree Deletion (1 child)

Removing a node with one child from a BST.

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Tree Deletion (2 children)

Removing a node with two children from a BST.

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BST

Binary Search Tree; a tree data structure where the value of each node is greater than the values of all nodes in its left subtree and less than the values of all nodes in its right subtree.

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Inorder Traversal

Tree traversal where you visit the left subtree, then the node, then the right subtree. (Left, Node, Right).

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Preorder Traversal

Tree traversal where you visit the node, then the left subtree, then the right subtree. (Node, Left, Right).

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Postorder Traversal

Tree traversal where you visit the left subtree, then the right subtree, then the node. (Left, Right, Node).

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Tree-Delete-Without-Parent

Algorithm to delete a node from a BST, without needing to track the parent node.

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Delete Node

Removing a specific node from a binary search tree

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Inorder Successor

Smallest node greater than a given node in a BST.

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TREE-MINIMUM

Recursive function to find the smallest node in a binary search tree.

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Node Deletion

Deleting a node from BST involves considering cases—no children, one child, or two children for a node.

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Key Value

Unique value associated with each node in BST, used for comparison.

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Tree Traversal

Systematically visiting every node in a tree.

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Binary Search Tree (BST) Structure

Nodes with left and right children where left is smaller and right is larger than the node's key value, ensuring orderliness

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Deleting in BST

Node removal following specific rules dependent on the node's children (none, one, two).

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Creating a BST

Inserting node values in a BST while maintaining the property that the left child is smaller and the right child is greater.

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BST-deletion algorithms

Specific algorithms are used to handle the removal of nodes based on their children.

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Binary Search Tree (BST)

A tree data structure where the value of each node is greater than the value of all nodes in its left subtree, and less than the value of all nodes in its right subtree.

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BST Properties

Left subtree keys are smaller, right subtree keys are larger than the current node's key, and both subtrees are also BSTs.

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BST Lookup Time

An average time complexity of O(log n) to search, insert, and delete— each comparison skips half the tree.

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BST Worst Case Height

In the worst case, a BST can have a height of n (number of nodes), becoming linear.

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Preorder Traversal

Visit root, then left, then right subtrees in the algorithm.

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Inorder Traversal

Visit left subtree, then root, then right subtree in a BST; this preserves the sorted order of nodes.

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Postorder Traversal

Visit left subtree, right subtree, then root in a BST.

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BST Node Structure

Nodes in a BST typically include data, pointers to the left and right child nodes.

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BST Search Time Complexity

The time to find a node in a BST is proportional to the tree's height.

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BST Insertion

Adding a node while preserving the BST property.

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Study Notes

Binary Search Tree (BST)

  • BSTs, sometimes called ordered or sorted binary trees, store keys in sorted order.
  • Properties:
    • The left subtree of a node contains only nodes with keys less than the node's key.
    • The right subtree of a node contains only nodes with keys greater than the node's key.
    • The left and right subtrees each must also be a binary search tree.
    • There must be no duplicate nodes.
  • Operations (lookup, addition, deletion) can be executed efficiently.
  • Operations traverse the tree from root to leaf, making comparisons to keys, and deciding whether to continue searching in left or right subtrees.
  • On average, each comparison allows operations to skip about half of the tree, making the time proportional to the logarithm of the number of items stored.
  • Average height of a BST with n nodes is O(log n).
  • Worst case height of a BST with n nodes is O(n).
  • A tree node has:
    • data: Data element
    • left: Pointer to left node
    • right: Pointer to right node
    • root: Pointer to root node (initially NULL)
  • newnode(int element) creates a new node in the tree.

Traversals

  • Preorder: (parent, left, right)
  • Inorder: (left, parent, right)
  • Postorder: (left, right, parent)
  • Example Preorder traversal: 60, 41, 16, 25, 53, 46, 42, 55, 74 …
  • Example Inorder traversal: 16, 25, 41, 42, 46, 53, 55, 60 …
  • Example Postorder traversal: 25, 16, 42, 46, 55, 53, 41, 62…

Searching

  • TREE-SEARCH(x, k):
    • If x is nil or k equals x.key, return x.
    • If k < x.key, return TREE-SEARCH(x.left, k).
    • Otherwise, return TREE-SEARCH(x.right, k).
  • ITERATIVE-TREE-SEARCH(x, k):
    • While x is not nil and k is not equal to x.key:
      • If k < x.key, x = x.left.
      • Otherwise, x = x.right.
    • Return x.
  • Running time (both iterative and recursive) is O(h), where h is the height of the tree.

Minimum and Maximum

  • TREE-MINIMUM(x):
    • While x.left is not nil, x = x.left.
    • Return x.
  • TREE-MAXIMUM(x):
    • While x.right is not nil, x = x.right.
    • Return x.

Successor and Predecessor

  • Successor: next node in inorder traversal.
  • Predecessor: previous node in inorder traversal.

Insertion

  • New key is inserted at a leaf node.
  • Search from root to leaf to find insertion point.
  • Add the new node as a child of the leaf node.
  • Duplicate keys are not allowed.
  • If a key already exists, return the existing node.

Deletion

  • Search for the node to remove.
  • Execute the removal algorithm.
    • Case 1: No children: Set parent's corresponding link to nil and dispose of the node.
    • Case 2: One child: Link the single child directly to the parent of the removed node.
    • Case 3: Two children: Find the inorder successor (minimum in the right subtree), copy its contents to the node being removed, and delete the inorder successor from the right subtree.

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Binary Search Tree PDF

Description

This quiz explores the fundamentals of Binary Search Trees (BSTs), including their properties, operations, and efficiency. Learn how BSTs organize data and the implications for searching, adding, and deleting nodes. Test your understanding of these essential data structures in computer science.

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