r
/
notebook
1
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity
notebook/notes/data-structures/binary-tree.md

13 KiB
Raw Blame History

title TARGET DECK FILE TAGS tags
Binary Tree Obsidian::STEM data_structure::bst
data_structure
graph
tree

Overview

A binary tree T is a structure defined on a finite set of nodes that either

  • contains no nodes, or
  • is composed of three disjoint sets of nodes: a root node, a left subtree, and a right subtree.

The binary tree is a specialization of the trees#Positional Trees.

%%ANKI Basic Is a binary tree a k-ary tree? Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is a binary tree a positional tree? Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%%

%%ANKI Basic Is a binary tree an ordered tree? Back: No. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for a binary tree to be full? Back: Each node has 0 or 2 children. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for a binary tree to be perfect? Back: Each leaf has the same depth and all internal nodes have degree 2. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is a perfect binary tree considered full? Back: Yes. Reference: “Binary Tree,” in Wikipedia, March 13, 2024, https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees.

END%%

%%ANKI Basic Is a full binary tree considered perfect? Back: Not necessarily. Reference: “Binary Tree,” in Wikipedia, March 13, 2024, https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees.

END%%

%%ANKI Basic Is a full binary tree considered complete? Back: Not necessarily. Reference: “Binary Tree,” in Wikipedia, March 13, 2024, https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees.

END%%

%%ANKI Basic Is a complete binary tree considered full? Back: Not necessarily. Reference: “Binary Tree,” in Wikipedia, March 13, 2024, https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees.

END%%

%%ANKI Basic What alternative term is sometimes used in favor of a "perfect binary tree"? Back: A "complete binary tree". Reference: “Binary Tree,” in Wikipedia, March 13, 2024, https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees.

END%% %%ANKI Basic What alternative term is sometimes used in favor over a "complete binary tree"? Back: Some authors may say "nearly complete" if the last level isn't completely filled. Reference: “Binary Tree,” in Wikipedia, March 13, 2024, https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees.

END%%

%%ANKI Basic What degrees are permitted in a full binary tree? Back: 0 or 2 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What degrees are permitted in a perfect binary tree? Back: 0 or 2 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What category of rooted tree does a binary tree fall under? Back: A positional tree or k-ary tree. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is a binary tree a positional tree? Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How many nodes are in a perfect binary tree of height h? Back: 2^{h+1} - 1 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: algebra::sequence

END%%

%%ANKI Basic How many internal nodes are in a perfect binary tree of height h? Back: 2^h - 1 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: algebra::sequence

END%%

%%ANKI Basic How many external nodes are in a perfect binary tree of height h? Back: 2^h Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: algebra::sequence

END%%

%%ANKI Basic How many nodes are on level d of a perfect binary tree? Back: 2^d Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: algebra::sequence

END%%

%%ANKI Basic How does the number of internal nodes compare to the number of external nodes in a perfect binary tree? Back: There is one more external node than internal node. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: algebra::sequence

END%%

%%ANKI Basic Is the following a perfect binary tree? !perfect-tree.png Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is the following a complete binary tree? !perfect-tree.png Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is the following a full binary tree? !perfect-tree.png Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is the following a perfect binary tree? !complete-tree.png Back: No. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is the following a complete binary tree? !complete-tree.png Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is the following a full binary tree? !complete-tree.png Back: No. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is the following a perfect binary tree? !non-complete-tree.png Back: No. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is the following a complete binary tree? !non-complete-tree.png Back: No. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is the following a full binary tree? !non-complete-tree.png Back: No. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the minimum number of nodes in a complete binary tree of height h? Back: 2^h Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the base case used in the recursive definition of a binary tree? Back: The empty set. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What recurrence is used in the recursive definition of a binary tree? Back: A binary tree is composed of a root node, a left subtree, and a right subtree. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How should the nil constructor of an inductive binary tree, say Tree, be defined? Back:

| nil : Tree α

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

END%%

%%ANKI Basic How should the non-nil constructor of an inductive binary tree, say Tree, be defined? Back:

| node : α → Tree α → Tree α → Tree α

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

END%%

%%ANKI Basic In the following binary tree type, what name is given to the first argument of node?

inductive Tree α where
| nil : Tree α
| node : α → Tree α → Tree α → Tree α

Back: The root node. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: lean

END%%

%%ANKI Basic In the following binary tree type, what name is given to the second argument of node?

inductive Tree α where
| nil : Tree α
| node : α → Tree α → Tree α → Tree α

Back: The left subtree. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: lean

END%%

%%ANKI Basic In the following binary tree type, what name is given to the third argument of node?

inductive Tree α where
| nil : Tree α
| node : α → Tree α → Tree α → Tree α

Back: The right subtree. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: lean

END%%

%%ANKI Basic Given the following binary tree implementation, how do you construct an empty tree?

inductive Tree α where
| nil : Tree α
| node : α → Tree α → Tree α → Tree α

Back: nil Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: lean

END%%

%%ANKI Basic Given the following binary tree implementation, how do you construct a tree with root a, left child b, and right child c?

inductive Tree α where
| nil : Tree α
| node : α → Tree α → Tree α → Tree α

Back: node 'a' (node 'b' nil nil) (node 'c' nil nil) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: lean

END%%

%%ANKI Basic Why isn't a binary tree considered an ordered tree? Back: A left child is distinct from a right child, even if the child is the same in both cases. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How many internal nodes are in a complete binary tree of n nodes? Back: \lceil (n - 1) / 2 \rceil = \lfloor n / 2 \rfloor Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic A node of a binary tree typically has what three pointers? Back: The parent, left child, and right child. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In what direction do binary trees grow? Back: New nodes are added to the bottom of the tree. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Bibliography