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Binary Search Tree Obsidian::STEM data_structure::bst
bst
data_structure

Overview

A binary search tree (BST) is a trees#Binary Trees satisfying the binary-search-tree property:

Let x be a node in a binary search tree. If y is a node in the left subtree of x, then y.key \leq x.key. If y is a node in the right subtree of x, then y.key \geq x.key.

Traversals

Consider an arbitrary node x of some BST. Then:

  • An inorder traversal visits x's left child, then x, then x's right child.
  • A preorder traversal visits x, then x's left child, then x's right child.
  • A postorder traversal visits x's left child, then x's right child, then x.

%%ANKI Basic Which of binary trees and binary search trees are more general? Back: Binary trees. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic A binary search tree is a binary tree with what property? Back: The binary-search-tree property. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does the binary-search-tree property state? Back: The value of a node is \geq those of its left subtree and \leq those of its right subtree. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let x be a binary search tree node and y be in x's left subtree. How do x and y compare? Back: y \leq x Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let x be a binary tree node and y be in x's right subtree. How do x and y compare? Back: Indeterminate. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let x be a binary search tree node and y be in x's right subtree. How do x and y compare? Back: x \leq y Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let x be a binary tree node and y be in x's left subtree. How do x and y compare? Back: Indeterminate. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In what order are nodes of a binary tree printed in an inorder traversal? Back: Left child, root, right child. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In what order are nodes of a binary tree printed in a postorder traversal? Back: Left child, right child, root. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In what order are nodes of a binary tree printed in a preorder traversal? Back: Root, left child, right child. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which binary tree node is printed first in an inorder traversal? Back: The leftmost child. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which binary tree node is printed last in an inorder traversal? Back: The rightmost child. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which binary tree node is printed first in a preorder traversal? Back: The root. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which binary tree node is printed last in a preorder traversal? Back: The rightmost child. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which binary tree node is printed first in a postorder traversal? Back: The leftmost child. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which binary tree node is printed last in a postorder traversal? Back: The root. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider the following binary search tree. List the nodes visited in postorder traversal. !binary-search-tree.png Back: 2, 5, 5, 8, 7, 6 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider the following binary search tree. List the nodes visited in preorder traversal. !binary-search-tree.png Back: 6, 5, 2, 5, 7, 8 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider the following binary search tree. List the nodes visited in inorder traversal. !binary-search-tree.png Back: 2, 5, 5, 6, 7, 8 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What path should be followed to find the minimum of a binary search tree? Back: The one formed by following all left pointers. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What path should be followed to find the maximum of a binary search tree? Back: The one formed by following all right pointers. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In terms of the height h of a BST, what is the runtime of search? Back: O(h) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In terms of the height h of a BST, what is the runtime for finding the minimum? Back: O(h) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In terms of the height h of a BST, what is the runtime for finding the maximum? Back: O(h) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Successors

The successor of a node in a binary search tree is the node whose value would appear immediately after in an in-order traversal.

%%ANKI Basic How do we define the successor of a BST node in terms of in-order traversals? Back: As the node encountered immediately after in an in-order traversal. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which node is the successor of 7 in the following BST? !binary-search-tree.png Back: 8 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which node is the successor of 2 in the following BST? !binary-search-tree.png Back: 5 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which node is the successor of 5 in the following BST? !binary-search-tree.png Back: Either 5 or 6. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which node(s) in a BST have no successor? Back: The maximum. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What two cases are considered when finding the successor of a BST node? Back: If the node has a right subtree or not. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Suppose a BST node has a right subtree. What is its successor? Back: The minimum of the right subtree. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Suppose a BST node does not have a right subtree. What is its successor? Back: The first proper ancestor reached from the LHS. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze If a BST node has a right subtree, it's successor cannot have a {left} subtree. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Suppose a BST node has a right subtree. Why can't its successor have a left subtree? Back: Because then a node in that left subtree would be the actual successor. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Suppose a BST node has a right subtree. Why can't its successor have a right subtree? Back: N/A. It can. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In terms of the height h of a BST, what is the runtime for finding a node's successor? Back: O(h) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Predecessors

The predecessor of a node in a binary search tree is the node whose value would appear immediately before in an in-order traversal.

%%ANKI Basic How do we define the predecessor of a BST node in terms of in-order traversals? Back: As the node encountered immediately before in an in-order traversal. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which node is the predecessor of 7 in the following BST? !binary-search-tree.png Back: 6 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which node is the predecessor of 2 in the following BST? !binary-search-tree.png Back: N/A. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which node is the predecessor of 5 in the following BST? !binary-search-tree.png Back: Either 2 or 5. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which node(s) in a BST have no predecessor? Back: The minimum. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What two cases are considered when finding the predecessor of a BST node? Back: If the node has a left subtree or not. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Suppose a BST node has a left subtree. What is its predecessor? Back: The maximum of the left subtree. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Suppose a BST node does not have a left subtree. What is its predecessor? Back: The first proper ancestor reached from the RHS. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze If a BST node has a left subtree, it's predecessor cannot have a {right} subtree. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Suppose a BST node has a left subtree. Why can't its successor have a left subtree? Back: N/A. It can. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Suppose a BST node has a left subtree. Why can't its predecessor have a right subtree? Back: Because then a node in that right subtree would be the actual predecessor. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In terms of the height h of a BST, what is the runtime for finding a node's predecessor? Back: O(h) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Deletions

Consider deleting node z from a BST. There are three conceptual cases to consider corresponding to the number of children z has:

  • If z has no children we can just replace z with NIL.
  • If z has one child, we can replace z with its child.
  • If z has two children, we swap z with either its predecessor or successor, updating pointers as necessary to maintain the binary-search-tree property.

%%ANKI Basic Insert node 7.5 into the following BST. Where is the new node located? !binary-search-tree.png Back: As 8's left child. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Insert node 6.5 into the following BST. Where is the new node located? !binary-search-tree.png Back: As 7's left child. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Insert node 5.5 into the following BST. Where is the new node located? !binary-search-tree.png Back: As the lower 5's right child. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic When does insertion into a BST modify the root node? Back: When the tree being inserted into is empty. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In terms of the height h of a BST, what is the runtime for inserting a node? Back: O(h) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How many cases are there to consider when deleting a node from a BST? Back: Three. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What corresponds to the cases to consider when deleting a node from a BST? Back: The number of children the node in question has. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Delete BST node z with no children. Which node is z replaced with? Back: NIL Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Delete BST node z with one child. Which node is z replaced with? Back: Its one child. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Delete BST node z with two children. Which node is z replaced with? Back: Either its successor or predecessor. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Delete BST node z with two children. If replacing with its successor, what two subcases need to be considered? Back: If z's successor is its right child or not. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Delete BST node z with two children. If replacing with its predecessor, what two subcases need to be considered? Back: If z's predessor is its left child or not. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In terms of the height h of a BST, what is the runtime for deleting a node? Back: O(h) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Delete z from the following BST. What does the resulting tree look like? !bst-right-child.png Back: !bst-right-child-after.png Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Delete z from the following BST. What does the resulting tree look like? !bst-left-child.png Back: !bst-left-child-after.png Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Delete z from the following BST. What does the resulting tree look like? !bst-right-succ.png Back: !bst-right-succ-after.png Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Delete z from the following BST. What does the resulting tree look like? !bst-deep-succ.png Back: !bst-deep-succ-after.png Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What arbitrary choice was implied in the deletion algorithm of the following BST? !bst-deep-succ.png Back: To replace deleted nodes with their successor instead of predecessor. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Bibliography

  • Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).