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B-trees, binary trees, and trichotomy.

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Joshua Potter 2024-08-09 17:15:58 -06:00
parent 618e016c1a
commit 3f1b3ec054
17 changed files with 946 additions and 496 deletions
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---
title: "2024-08-09"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
* First pass on [[b-tree|B-tree]] notes.
* Notes on [[relations#Trichotomy|trichotomy]].

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---
title: "2024-08-04"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
* Expanded notes on C [[c17/declarations|declarations]].

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---
title: "2024-08-04"
title: "2024-08-05"
---
- [x] Anki Flashcards

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---
title: "2024-08-06"
---
- [ ] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)

9
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---
title: "2024-08-07"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)

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---
title: "2024-08-08"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
* Reading on B-trees in "Introduction to Algorithms".

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@ -55,6 +55,22 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1717554445670-->
END%%
%%ANKI
Basic
Is the symmetric difference commutative?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187675-->
END%%
%%ANKI
Basic
Is the symmetric difference associative?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187680-->
END%%
## Cartesian Product
Given two sets $A$ and $B$, the **Cartesian product** $A \times B$ is defined as: $$A \times B = \{\langle x, y \rangle \mid x \in A \land y \in B\}$$

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---
title: B-Tree
TARGET DECK: Obsidian::STEM
FILE TAGS: data_structure::b-tree
tags:
- b-tree
- data_structure
---
## Overview
A **B-tree of order $m$** is a tree that satisfies the following properties:
* Every node has at most $m$ children.
* Every node, except for the root, has at least $m / 2$ children.
* All leaves appear on the same level.
* A node with $k$ children contains $k - 1$ keys sorted in monotonically increasing order.
The above is a modification of Knuth's definition in his "Art of Computer Programming" that defines leaves of the tree more consistently with how I use the term elsewhere. It also pulls in concepts from CLRS (such as keys needing to be sorted within nodes).
%%ANKI
Basic
What hyperparameter is used to define a B-tree?
Back: It's order, i.e. the maximum number of a children a node can have.
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211541967-->
END%%
%%ANKI
Basic
In what direction do B-trees grow?
Back: From bottom to top.
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211541998-->
END%%
%%ANKI
Basic
Consider B-tree of order $m$. What does $m$ refer to?
Back: The maximum number of children a node can have.
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542004-->
END%%
%%ANKI
Basic
What is the maximum number of children a node in a B-tree have?
Back: N/A. It depends on the tree's order.
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542010-->
END%%
%%ANKI
Basic
What is the maximum number of children a node in a B-tree of order $m$ can have?
Back: $m$
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542016-->
END%%
%%ANKI
Basic
What is the minimum number of children a non-root node in a B-tree of order $m$ can have?
Back: $\lceil m / 2 \rceil$
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542022-->
END%%
%%ANKI
Basic
What is the maximum number of keys a node in a B-tree of order $m$ can have?
Back: $m - 1$
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542028-->
END%%
%%ANKI
Basic
What is the minimum number of keys a non-root node in a B-tree can have?
Back: N/A. It depends on the tree's order.
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542035-->
END%%
%%ANKI
Basic
What is the minimum number of keys a non-root node in a B-tree of order $m$ can have?
Back: $\lceil m / 2 \rceil - 1$
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542041-->
END%%
%%ANKI
Basic
A node in a B-tree of order $m$ has $k$ keys. How many children does it have?
Back: $k + 1$
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542046-->
END%%
%%ANKI
Basic
A node in a B-tree of order $m$ has $k$ children. How many keys does it have?
Back: $k - 1$
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542052-->
END%%
%%ANKI
Basic
*When* does a B-tree gain height?
Back: When the root node is split.
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542058-->
END%%
%%ANKI
Basic
Consider a B-tree of order $7$. How many children $c$ can each non-root node have?
Back: $4 \leq c \leq 7$
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542063-->
END%%
%%ANKI
Basic
Consider a B-tree of order $7$. How many children $c$ can the root have?
Back: $0 \leq c \leq 7$
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542069-->
END%%
%%ANKI
Basic
Consider a B-tree of order $7$. How many keys $k$ can each non-root node have?
Back: $3 \leq k < 7$
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542076-->
END%%
%%ANKI
Basic
Consider a B-tree of order $7$. How many keys $k$ can the root have?
Back: $0 \leq k < 7$
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542082-->
END%%
%%ANKI
Basic
What instances exist of a B-tree of order $1$?
Back: Just the empty tree.
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542088-->
END%%
%%ANKI
Basic
*Why* can't we define a nonempty B-tree of order $1$?
Back: Each node can have at most $1$ child, meaning each node contains $0$ keys.
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542094-->
END%%
%%ANKI
Basic
How are keys arranged within a B-tree's nodes?
Back: In monotonically increasing order.
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542105-->
END%%
## Bibliography
* Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

19
notes/data-structures/binary-search-tree.md
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@ -13,6 +13,14 @@ A binary search tree (BST) is a [[trees#Binary Trees|binary tree]] satisfying th
> 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 $y.key \geq x.key$.
%%ANKI
Basic
*Why* can't we define a binary search tree as a B-tree of order $2$?
Back: A BST isn't guaranteed to be balanced.
Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
<!--ID: 1723211542110-->
END%%
## Traversals
Consider an arbitrary node $x$ of some BST. Then:
@ -455,14 +463,6 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
<!--ID: 1722713303038-->
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).
<!--ID: 1722713303039-->
END%%
%%ANKI
Basic
In terms of the height $h$ of a BST, what is the runtime for inserting a node?
@ -522,7 +522,7 @@ 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.
Back: If $z$'s predecessor is its left child or not.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303047-->
END%%
@ -586,4 +586,5 @@ END%%
## Bibliography
* Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

417
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@ -0,0 +1,417 @@
---
title: Binary Tree
TARGET DECK: Obsidian::STEM
FILE TAGS: data_structure::bst
tags:
- 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|k-ary tree]].
%%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).
<!--ID: 1714089436138-->
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).
<!--ID: 1714089436144-->
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).
<!--ID: 1713118128213-->
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).
<!--ID: 1714081594570-->
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](https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees).
<!--ID: 1714088438720-->
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](https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees).
<!--ID: 1714088438726-->
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](https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees).
<!--ID: 1714088438729-->
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](https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees).
<!--ID: 1714088438733-->
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](https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees).
<!--ID: 1714088438737-->
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](https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees).
<!--ID: 1714088438744-->
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).
<!--ID: 1714081594576-->
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).
<!--ID: 1714081594579-->
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).
<!--ID: 1714081594582-->
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).
<!--ID: 1713118128227-->
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
<!--ID: 1713118128255-->
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
<!--ID: 1714080353472-->
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
<!--ID: 1714080353469-->
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
<!--ID: 1714080353465-->
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
<!--ID: 1714080353476-->
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).
<!--ID: 1714180419777-->
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).
<!--ID: 1714180419781-->
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).
<!--ID: 1714180419784-->
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).
<!--ID: 1714180419787-->
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).
<!--ID: 1714180419789-->
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).
<!--ID: 1714180419793-->
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).
<!--ID: 1714180419802-->
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).
<!--ID: 1714180419809-->
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).
<!--ID: 1714180419813-->
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).
<!--ID: 1714082676010-->
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).
<!--ID: 1712409466593-->
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).
<!--ID: 1712409466606-->
END%%
%%ANKI
Basic
How should the nil constructor of an inductive binary tree, say `Tree`, be defined?
Back:
```lean
| nil : Tree α
```
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: lean
<!--ID: 1712409466615-->
END%%
%%ANKI
Basic
How should the non-nil constructor of an inductive binary tree, say `Tree`, be defined?
Back:
```lean
| 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
<!--ID: 1712409466621-->
END%%
%%ANKI
Basic
In the following binary tree type, what name is given to the first argument of `node`?
```lean
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
<!--ID: 1712409466627-->
END%%
%%ANKI
Basic
In the following binary tree type, what name is given to the second argument of `node`?
```lean
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
<!--ID: 1712409466634-->
END%%
%%ANKI
Basic
In the following binary tree type, what name is given to the third argument of `node`?
```lean
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
<!--ID: 1712409466639-->
END%%
%%ANKI
Basic
Given the following binary tree implementation, how do you construct an empty tree?
```lean
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
<!--ID: 1712409466643-->
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`?
```lean
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
<!--ID: 1712409466648-->
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).
<!--ID: 1712409466653-->
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).
<!--ID: 1714349367662-->
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).
<!--ID: 1715969047059-->
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).
<!--ID: 1723208264398-->
END%%
## Bibliography
* “Binary Tree,” in _Wikipedia_, March 13, 2024, [https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees](https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees).
* Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

2
notes/hashing/addressing.md
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@ -555,7 +555,7 @@ END%%
%%ANKI
Basic
When is the load factor of an open addressing hash table $> 1$?
Back: N/A
Back: N/A.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1718759188186-->
END%%

2
notes/hashing/index.md
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@ -585,7 +585,7 @@ END%%
%%ANKI
Basic
Consider universal family $\mathscr{H}$ and universe $U$. What does the following evaluate to? $$|\{h \in \mathscr{H} \mid h(x) = h(y)\}| \text{ for distinct } x, y \in U$$
Consider universal family $\mathscr{H}$ and universe $U$. What number does the following evaluate to? $$|\{h \in \mathscr{H} \mid h(x) = h(y)\}| \text{ for distinct } x, y \in U$$
Back: A value between $0$ and $|\mathscr{H}|$ inclusive.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: hashing::random hashing::universal

2
notes/programming/pred-trans.md
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@ -971,7 +971,7 @@ END%%
%%ANKI
Basic
When is the first guarded command of the following executed? $$\begin{align*} \textbf{if } & x \geq 0 \rightarrow z \coloneqq x \\ \textbf{ | } & x \leq 0 \rightarrow z \coloneqq -x \\ \textbf{fi } & \end{align*}$$
Back: When $x \geq 0$.
Back: When $x > 0$ or (possibly) when $x = 0$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722257348955-->
END%%

2
notes/set/index.md
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@ -269,7 +269,7 @@ END%%
%%ANKI
Cloze
$P(x)$ is equivalently written as $x \in$ {$\{v \mid P(v)\}$}.
$P(x) = T$ is equivalently written as $x \in$ {$\{v \mid P(v)\}$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720369624733-->
END%%

144
notes/set/relations.md
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@ -1008,7 +1008,7 @@ Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https:
END%%
%%ANKI
Give an example of a relation that is both symmetric and asymmetric?
Give an example of a relation that is both symmetric and asymmetric.
Back: $\varnothing$
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
END%%
@ -1030,9 +1030,11 @@ Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https:
END%%
%%ANKI
Give an example of a nonempty relation that is both symmetric and asymmetric?
Basic
Give an example of a nonempty relation that is both symmetric and asymmetric.
Back: N/A.
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
<!--ID: 1723245187584-->
END%%
%%ANKI
@ -1098,6 +1100,13 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1722735199608-->
END%%
%%ANKI
Cloze
A relation $R$ is asymmetric iff $R$ is {antisymmetric} and {irreflexive}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187594-->
END%%
## Transitivity
A relation $R$ is **transitive** iff whenever $xRy$ and $yRz$, then $xRz$. In relational algebra, we define $R$ to be transitive iff $R \circ R \subseteq R$.
@ -1272,6 +1281,137 @@ Reference: “Connected Relation,” in _Wikipedia_, July 14, 2024, [https://en.
<!--ID: 1722735199715-->
END%%
## Trichotomy
A binary relation $R$ on $A$ is **trichotomous** if for all $x, y \in A$, exactly one of the following holds: $$xRy, \quad x = y, \quad yRx$$
%%ANKI
Basic
How is trichotomy of relation $R$ on set $A$ defined in FOL?
Back: $\forall x, y \in A, (xRy \land x \neq y \land \neg yRx) \lor (\neg xRy \land x = y \land \neg yRx) \lor (\neg xRy \land x \neq y \land yRx)$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187598-->
END%%
%%ANKI
Basic
Is $R = \{\langle 2, 3 \rangle, \langle 2, 5 \rangle, \langle 3, 5 \rangle\}$ trichotomous on $\{2, 3, 5\}$?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187602-->
END%%
%%ANKI
Basic
Is $R = \{\langle 2, 3 \rangle, \langle 3, 5 \rangle\}$ trichotomous on $\{2, 3, 5\}$?
Back: No.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187605-->
END%%
%%ANKI
Basic
*Why* isn't $R = \{\langle 2, 3 \rangle, \langle 3, 5 \rangle\}$ trichotomous on $\{2, 3, 5\}$?
Back: Because no ordered pair relates $2$ and $5$ together.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187609-->
END%%
%%ANKI
Basic
Is $R = \{\langle 2, 2 \rangle\}$ trichotomous on $\{2\}$?
Back: No.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187613-->
END%%
%%ANKI
Basic
*Why* isn't $R = \{\langle a, a \rangle\}$ trichotomous on $\{a\}$?
Back: Because $aRa$ and $a = a$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187617-->
END%%
%%ANKI
Basic
Can a relation be both reflexive and trichotomous?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187621-->
END%%
%%ANKI
Basic
Can a nonempty relation be both reflexive and trichotomous?
Back: No.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187628-->
END%%
%%ANKI
Basic
Can a nonempty relation be both irreflexive and trichotomous?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187633-->
END%%
%%ANKI
Basic
Which of trichotomy or irreflexivity is more general?
Back: Irreflexivity.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187638-->
END%%
%%ANKI
Basic
*Why* must trichotomous relations on (say) set $A$ be irreflexive?
Back: For any $x \in A$, it follows $x = x$. Then $\neg xRx$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187643-->
END%%
%%ANKI
Basic
Can a nonempty relation be both symmetric and trichotomous?
Back: No.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187648-->
END%%
%%ANKI
Basic
Can a nonempty relation be both antisymmetric and trichotomous?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187654-->
END%%
%%ANKI
Basic
Which of antisymmetry or trichotomy is more general?
Back: Antisymmetry.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187659-->
END%%
%%ANKI
Basic
*Why* must trichotomous relations on (say) set $A$ be antisymmetric?
Back: For any $x, y \in A$, if $x \neq y$ then $xRy$ or $yRx$ but not both.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187664-->
END%%
%%ANKI
Cloze
A relation $R$ is trichotomous iff $R$ is {asymmetric} and {connected}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1723245187669-->
END%%
## Equivalence Relations
Given relation $R$ and set $A$, $R$ is an **equivalence relation on $A$** iff $R$ is a binary relation on $A$ that is reflexive on $A$, symmetric, and transitive.

398
notes/set/trees.md
View File

@ -55,7 +55,7 @@ END%%
%%ANKI
Basic
What additional properties must a free tree exhibit to be a forest?
Back: N/A
Back: N/A.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1711136844915-->
END%%
@ -852,7 +852,7 @@ END%%
### Positional Trees
A **positional tree** is a rooted tree in which each child is labeled with a specific positive integer. A **$k$-ary tree** is a positional tree with at most $k$ children/labels. A binary tree is a $2$-ary tree.
A **positional tree** is a rooted tree in which each child is labeled with a specific positive integer. A **$k$-ary tree** is a positional tree with at most $k$ children/labels. A [[binary-tree|binary tree]] is a $2$-ary tree.
A $k$-ary tree is **full** if every node has degree $0$ or $k$. A $k$-ary tree is **perfect** if all leaves have the same depth and all internal nodes have degree $k$. A $k$-ary tree is **complete** if the last level is not filled but all leaves have the same depth and are leftmost arranged.
@ -1197,400 +1197,6 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
<!--ID: 1714349367655-->
END%%
#### Binary Trees
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**.
%%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).
<!--ID: 1714089436138-->
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).
<!--ID: 1714089436144-->
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).
<!--ID: 1713118128213-->
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).
<!--ID: 1714081594570-->
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](https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees).
<!--ID: 1714088438720-->
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](https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees).
<!--ID: 1714088438726-->
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](https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees).
<!--ID: 1714088438729-->
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](https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees).
<!--ID: 1714088438733-->
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](https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees).
<!--ID: 1714088438737-->
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](https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees).
<!--ID: 1714088438744-->
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).
<!--ID: 1714081594576-->
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).
<!--ID: 1714081594579-->
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).
<!--ID: 1714081594582-->
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).
<!--ID: 1713118128227-->
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
<!--ID: 1713118128255-->
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
<!--ID: 1714080353472-->
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
<!--ID: 1714080353469-->
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
<!--ID: 1714080353465-->
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
<!--ID: 1714080353476-->
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).
<!--ID: 1714180419777-->
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).
<!--ID: 1714180419781-->
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).
<!--ID: 1714180419784-->
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).
<!--ID: 1714180419787-->
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).
<!--ID: 1714180419789-->
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).
<!--ID: 1714180419793-->
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).
<!--ID: 1714180419802-->
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).
<!--ID: 1714180419809-->
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).
<!--ID: 1714180419813-->
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).
<!--ID: 1714082676010-->
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).
<!--ID: 1712409466593-->
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).
<!--ID: 1712409466606-->
END%%
%%ANKI
Basic
How should the nil constructor of an inductive binary tree, say `Tree`, be defined?
Back:
```lean
| nil : Tree α
```
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: lean
<!--ID: 1712409466615-->
END%%
%%ANKI
Basic
How should the non-nil constructor of an inductive binary tree, say `Tree`, be defined?
Back:
```lean
| 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
<!--ID: 1712409466621-->
END%%
%%ANKI
Basic
In the following binary tree type, what name is given to the first argument of `node`?
```lean
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
<!--ID: 1712409466627-->
END%%
%%ANKI
Basic
In the following binary tree type, what name is given to the second argument of `node`?
```lean
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
<!--ID: 1712409466634-->
END%%
%%ANKI
Basic
In the following binary tree type, what name is given to the third argument of `node`?
```lean
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
<!--ID: 1712409466639-->
END%%
%%ANKI
Basic
Given the following binary tree implementation, how do you construct an empty tree?
```lean
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
<!--ID: 1712409466643-->
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`?
```lean
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
<!--ID: 1712409466648-->
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).
<!--ID: 1712409466653-->
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).
<!--ID: 1714349367662-->
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).
<!--ID: 1715969047059-->
END%%
## Bibliography
* “Binary Tree,” in _Wikipedia_, March 13, 2024, [https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees](https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees).
* Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).