Heaps and science of programming notes.

notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json

@@ -114,7 +114,9 @@
     "nearly-complete-tree.png",
     "non-nearly-complete-tree.png",
     "perfect-tree.png",
-    "non-complete-tree.png"
+    "non-complete-tree.png",
+    "max-heap-tree.png",
+    "max-heap-array.png"
   ],
   "File Hashes": {
     "algorithms/index.md": "3ac071354e55242919cc574eb43de6f8",
@@ -153,7 +155,7 @@
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-    "logic/equiv-trans.md": "5d68be4120e7f79e12c35fafeddaf1a1",
+    "logic/equiv-trans.md": "63515dfd471653bc0c8bc4eca66ccaf8",
     "_journal/2024-02-07.md": "8d81cd56a3b33883a7706d32e77b5889",
     "algorithms/loop-invariants.md": "cbefc346842c21a6cce5c5edce451eb2",
     "algorithms/loop-invariant.md": "3b390e720f3b2a98e611b49a0bb1f5a9",
@@ -201,7 +203,7 @@
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     "_journal/2024-02/2024-02-15.md": "16cb7563d404cb543719b7bb5037aeed",
-    "algebra/floor-ceiling.md": "ffffdd893398da842be2e4337c16bbf7",
+    "algebra/floor-ceiling.md": "412e4fdc424cb17fd818688857c4b5b3",
     "algebra/index.md": "90b842eb694938d87c7c68779a5cacd1",
     "algorithms/binary-search.md": "8533a05ea372e007ab4e8a36fd2772a9",
     "_journal/2024-02-17.md": "7c37cb10515ed3d2f5388eaf02a67048",
@@ -245,7 +247,7 @@
     "filesystems/cas.md": "d41c0d2e943adecbadd10a03fd1e4274",
     "git/objects.md": "c6b7e6a26666386790d25d4ece38175d",
     "git/index.md": "83d2d95fc549d9e8436946c7bd058d15",
-    "encoding/integer.md": "73c42283ff8c3a1d4efb9f9644a4cf3b",
+    "encoding/integer.md": "2ab21152b9468f547ebee496e03bc410",
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     "_journal/2024-02/2024-02-28.md": "7489377c014a2ff3c535d581961b5b82",
     "_journal/2024-03-01.md": "a532486279190b0c12954966cbf8c3fe",
@@ -315,7 +317,7 @@
     "x86-64/declarations.md": "75bc7857cf2207a40cd7f0ee056af2f2",
     "x86-64/instructions.md": "d783bad8dd77748fb412715541cb844d",
     "git/refs.md": "954fc69004aa65b358ec5ce07c1435ce",
-    "set/trees.md": "6c1a245f6b25838c519f5c9bcf96c369",
+    "set/trees.md": "0d21b947917498f107da140cc9fb93a7",
     "_journal/2024-03-24.md": "1974cdb9fc42c3a8bebb8ac76d4b1fd6",
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@@ -371,7 +373,19 @@
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     "_journal/2024-04/2024-04-24.md": "4cb04e0dea56e0b471fc0e428471a390",
-    "algorithms/heaps.md": "4235ec90b6e251fc992906e0f1fe69e7"
+    "algorithms/heaps.md": "b12c70ec85e514ce912821d133d116d4",
+    "_journal/2024-04-26.md": "3ce37236a9e09e74b547a4f7231df5f0",
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+    "_journal/2024-04/2024-04-26.md": "3ce37236a9e09e74b547a4f7231df5f0",
+    "algorithms/sorting/heapsort.md": "e0d368882a8f33d2f42bd3ead4e3914d",
+    "_journal/2024-04-29.md": "7888f4e9497c9d8bd6f4aa759d9abc4d",
+    "_journal/2024-04/2024-04-28.md": "b34a9fe3bccb1f224b96ca00e78ad061",
+    "programming/assertions.md": "bdef9b934d8db94169d6befbc02f33d2",
+    "programming/text-sub.md": "003b8c32ae2f6767cb0d900f85196e67",
+    "programming/equiv-trans.md": "fbadd593688e364ea92f74f23e54bcfc",
+    "programming/index.md": "bb082325e269a95236aa6aff9307fe59"
   },
   "fields_dict": {
     "Basic": [

notes/_journal/2024-04-29.md

@@ -0,0 +1,14 @@
+---
+title: "2024-04-29"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] Sheet Music (10 min.)
+- [ ] Go (1 Life & Death Problem)
+- [ ] Korean (Read 1 Story)
+- [ ] Interview Prep (1 Practice Problem)
+- [x] Log Work Hours (Max 3 hours)
+
+* Notes on chapter 5.3 of "The Science of Programming". Covered nested arrays.
+* Read chapter 6 of "The Science of Programming". Still need to convert into notes though.

notes/_journal/2024-04/2024-04-26.md

@@ -0,0 +1,11 @@
+---
+title: "2024-04-26"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] Sheet Music (10 min.)
+- [ ] Go (1 Life & Death Problem)
+- [ ] Korean (Read 1 Story)
+- [ ] Interview Prep (1 Practice Problem)
+- [x] Log Work Hours (Max 3 hours)

notes/_journal/2024-04/2024-04-27.md

@@ -0,0 +1,14 @@
+---
+title: "2024-04-27"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] Sheet Music (10 min.)
+- [ ] Go (1 Life & Death Problem)
+- [ ] Korean (Read 1 Story)
+- [ ] Interview Prep (1 Practice Problem)
+- [ ] Log Work Hours (Max 3 hours)
+
+* Hide-and-Seek Application
+	* Finished most game logic. Began working on theming.

notes/_journal/2024-04/2024-04-28.md

@@ -0,0 +1,18 @@
+---
+title: "2024-04-28"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] Sheet Music (10 min.)
+- [ ] Go (1 Life & Death Problem)
+- [ ] Korean (Read 1 Story)
+- [ ] Interview Prep (1 Practice Problem)
+- [x] Log Work Hours (Max 3 hours)
+
+* Hide-and-Seek Application
+	* Added booting and teardown game server logic.
+	* Begin theming according to Fort Collins vendor guidelines.
+* Notes on chapter 5.2 in "The Science of Programming".
+* Notes on floor/ceiling identities associated with complete $k$-ary trees.
+* Start adding notes/flashcards on heaps and heapsort.

notes/algebra/floor-ceiling.md

@@ -157,7 +157,7 @@ END%%
 %%ANKI
 Basic
 What C operator corresponds to ceiling division?
-Back: None.
+Back: N/A.
 Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
 <!--ID: 1708110779716-->
 END%%
@@ -352,6 +352,136 @@ Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete M
 <!--ID: 1708115683366-->
 END%%
 
+## Identities
+
+For integers $x$ and $y > 0$, $$\begin{align*} \left\lfloor \frac{x}{y} \right\rfloor & = \left\lceil \frac{x}{y} - \frac{y - 1}{y} \right\rceil \\ \left\lceil \frac{x}{y} \right\rceil & = \left\lfloor \frac{x}{y} + \frac{y - 1}{y} \right\rfloor \end{align*}$$
+
+%%ANKI
+Basic
+If $n$ is even, what integer value does $\lfloor n / 2 \rfloor$ evaluate to?
+Back: $n / 2$
+Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
+<!--ID: 1714182124789-->
+END%%
+
+%%ANKI
+Basic
+If $n$ is odd, what integer value does $\lfloor n / 2 \rfloor$ evaluate to?
+Back: $(n - 1) / 2$
+Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
+<!--ID: 1714182124796-->
+END%%
+
+%%ANKI
+Basic
+If $n$ is even, what integer value does $\lceil n / 2 \rceil$ evaluate to?
+Back: $n / 2$
+Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
+<!--ID: 1714182124804-->
+END%%
+
+%%ANKI
+Basic
+If $n$ is odd, what integer value does $\lceil n / 2 \rceil$ evaluate to?
+Back: $(n + 1) / 2$
+Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
+<!--ID: 1714182124809-->
+END%%
+
+%%ANKI
+Basic
+Given integers $x$ and $y > 0$, what value of $Bias$ satisfies the following identity? $$\left\lceil \frac{x}{y} \right\rceil = \left\lfloor \frac{x}{y} + Bias \right\rfloor$$
+Back: $(y - 1) / y$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1714182124840-->
+END%%
+
+%%ANKI
+Basic
+Given integers $x$ and $y > 0$, what value of $Bias$ satisfies the following identity? $$\left\lceil \frac{x}{y} \right\rceil = \left\lfloor \frac{x + Bias}{y} \right\rfloor$$
+Back: $(y - 1)$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1714184300367-->
+END%%
+
+%%ANKI
+Basic
+Given integers $x$ and $y > 0$, what operator satisfies the following identity? $$\left\lceil \frac{x}{y} \right\rceil = \left\lfloor \frac{x}{y} \;\square\; \frac{y - 1}{y} \right\rfloor$$
+Back: $+$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1714182124853-->
+END%%
+
+%%ANKI
+Basic
+What intuition explains why the following identity holds for integers $x$ and $y > 0$? $$\left\lceil \frac{x}{y} \right\rceil = \left\lfloor \frac{x}{y} + \frac{y - 1}{y} \right\rfloor$$
+Back: $(y - 1) / y$ only affects the RHS if and only if $x / y$ is not an integer.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1714182124860-->
+END%%
+
+%%ANKI
+Basic
+Given integers $x$ and $y > 0$, what value of $Bias$ satisfies the following identity? $$\left\lfloor \frac{x}{y} \right\rfloor = \left\lceil \frac{x}{y} - Bias \right\rceil$$
+Back: $(y - 1) / y$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1714182124874-->
+END%%
+
+%%ANKI
+Basic
+Given integers $x$ and $y > 0$, what value of $Bias$ satisfies the following identity? $$\left\lfloor \frac{x}{y} \right\rfloor = \left\lceil \frac{x - Bias}{y} \right\rceil$$
+Back: $(y - 1)$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1714184300372-->
+END%%
+
+%%ANKI
+Basic
+Given integers $x$ and $y > 0$, what operator satisfies the following identity? $$\left\lfloor \frac{x}{y} \right\rfloor = \left\lceil \frac{x}{y} \;\square\; \frac{y - 1}{y} \right\rceil$$
+Back: $-$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1714182124867-->
+END%%
+
+%%ANKI
+Basic
+What intuition explains why the following identity holds for integers $x$ and $y > 0$? $$\left\lfloor \frac{x}{y} \right\rfloor = \left\lceil \frac{x}{y} - \frac{y - 1}{y} \right\rceil$$
+Back: $(y - 1) / y$ only affects the RHS if and only if $x / y$ is not an integer.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1714182124884-->
+END%%
+
+%%ANKI
+Cloze
+For any integer $n$, floor expression {$\lfloor n / 2 \rfloor$} is equal to ceiling expression {$\lceil (n - 1) / 2 \rceil$}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1714349367669-->
+END%%
+
+%%ANKI
+Cloze
+For any integer $n$, ceiling expression {$\lceil n / 2 \rceil$} is equal to floor expression {$\lfloor (n + 1) / 2 \rfloor$}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1714349367676-->
+END%%
+
+%%ANKI
+Basic
+What identity generalizes the following? $$\left\lfloor \frac{n}{2} \right\rfloor = \left\lceil \frac{n - 1}{2} \right\rceil$$
+Back: $$\left\lfloor \frac{n}{d} \right\rfloor = \left\lceil \frac{n - (d - 1)}{d} \right\rceil$$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1714349367682-->
+END%%
+
+%%ANKI
+Basic
+What identity generalizes the following? $$\left\lceil \frac{n}{2} \right\rceil = \left\lfloor \frac{n + 1}{2} \right\rfloor$$
+Back: $$\left\lceil \frac{n}{d} \right\rceil = \left\lfloor \frac{n + (d - 1)}{d} \right\rfloor$$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1714349367688-->
+END%%
+
 ## Bibliography
 
 * Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.

notes/algorithms/heaps.md

@@ -9,4 +9,144 @@ tags:
 
 ## Overview
 
-TODO
+The **binary heap** data structure is an array object that can be viewed as a [[trees#Positional Trees|complete binary tree]].
+
+%%ANKI
+Cloze
+A binary heap is an {array} that can be viewed as a {binary tree}.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714356379014-->
+END%%
+
+%%ANKI
+Basic
+Is the following a valid binary heap?
+![[perfect-tree.png]]
+Back: Yes.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714356379021-->
+END%%
+
+%%ANKI
+Basic
+Is the following a valid binary heap?
+![[complete-tree.png]]
+Back: Yes.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714356379024-->
+END%%
+
+%%ANKI
+Basic
+Is the following a valid binary heap?
+![[non-complete-tree.png]]
+Back: No.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714356379030-->
+END%%
+
+%%ANKI
+Basic
+Why can't the following be a binary heap?
+![[non-complete-tree.png]]
+Back: A heap is equivalently viewed as a *complete* binary tree.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714356379034-->
+END%%
+
+%%ANKI
+Basic
+What two sizes are associated with binary heaps?
+Back: The number of valid elements and the size of the underlying array.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714356379038-->
+END%%
+
+%%ANKI
+Basic
+What are the two kinds of binary heaps?
+Back: Max-heaps and min-heaps.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714356379042-->
+END%%
+
+%%ANKI
+Basic
+What is the max-heap property?
+Back: Every parent node is greater than or equal to its children in value.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714356379046-->
+END%%
+
+%%ANKI
+Basic
+Where is the largest element of a max-heap?
+Back: At the root.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714356379052-->
+END%%
+
+%%ANKI
+Basic
+Where is the smallest element of a max-heap?
+Back: At the leaves.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714356379059-->
+END%%
+
+%%ANKI
+Basic
+What is the min-heap property?
+Back: Every parent node is less than or equal to its children in value.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714356379072-->
+END%%
+
+%%ANKI
+Basic
+Where is the smallest element of a min-heap?
+Back: At the root.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714356379077-->
+END%%
+
+%%ANKI
+Basic
+Where is the largest element of a min-heap?
+Back: At the leaves.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714356379083-->
+END%%
+
+%%ANKI
+Basic
+How is the following binary heap viewed as an array?
+![[max-heap-tree.png]]
+Back:
+![[max-heap-array.png]]
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714356379065-->
+END%%
+
+%%ANKI
+Basic
+How is the following binary heap instead viewed as a binary tree?
+![[max-heap-array.png]]
+Back:
+![[max-heap-tree.png]]
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714356442370-->
+END%%
+
+%%ANKI
+Basic
+What kind of binary heap is the following?
+![[max-heap-array.png]]
+Back: A max-heap.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714356546616-->
+END%%
+
+## Bibliography
+
+* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

notes/algorithms/sorting/heapsort.md

@@ -0,0 +1,64 @@
+---
+title: Heapsort
+TARGET DECK: Obsidian::STEM
+FILE TAGS: algorithm::sorting
+tags:
+  - algorithm
+  - sorting
+---
+
+## Overview
+
+Property    | Value
+----------- | --------
+Best Case   | -
+Worst Case  | -
+Avg. Case   | -
+Aux. Memory | -
+Stable      | -
+Adaptive    | -
+
+![[heapsort.gif]]
+
+```c
+inline int left_child(int i) { return (i << 1) + 1; }
+inline int right_child(int i) { return (i << 1) + 2; }
+
+void max_heapify(int n, int H[static n], int i) {
+  while (true) {
+    int lc = left_child(i);
+    int rc = right_child(i);
+    int next = i;
+
+    if (lc < n && H[next] < H[lc]) {
+      next = lc;
+    }
+    if (rc < n && H[next] < H[rc]) {
+      next = rc;
+    }
+    if (next == i) {
+      return;
+    }
+    swap(H, i, next);
+    i = next;
+  }
+}
+
+void build_max_heap(int n, int H[static n]) {
+  for (int i = n / 2 - 1; i >= 0; --i) {
+    max_heapify(n, H, i);
+  }
+}
+
+void heapsort(int n, int H[static n]) {
+  build_max_heap(n, H);
+  while (n > 1) {
+    swap(A, 0, --n);
+    max_heapify(n, A, 0);
+  }
+}
+```
+
+## Bibliography
+
+* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

notes/encoding/integer.md

@@ -2114,6 +2114,67 @@ Tags: c17
 <!--ID: 1709831032461-->
 END%%
 
+%%ANKI
+Basic
+Assuming no overflow, rewrite expression `x >> k` to instead yield $\lceil x / 2^k \rceil$.
+Back: `(x + (1 << k) - 1) >> k`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c17
+<!--ID: 1714184300343-->
+END%%
+
+%%ANKI
+Basic
+Assuming no overflow, what is the result of `(x + (1 << k) - 1) >> k`?
+Back: $\lceil x / 2^k \rceil$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c17
+<!--ID: 1714184300349-->
+END%%
+
+%%ANKI
+Basic
+What value of $Bias$ satisfies the following identity? $$\left\lceil \frac{x}{2^k} \right\rceil = \left\lfloor \frac{x}{2^k} + Bias \right\rfloor$$
+Back: $(2^k - 1) / 2^k$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1714184300352-->
+END%%
+
+%%ANKI
+Basic
+What value of $Bias$ satisfies the following identity? $$\left\lceil \frac{x}{2^k} \right\rceil = \left\lfloor \frac{x + Bias}{2^k} \right\rfloor$$
+Back: $2^k - 1$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1714184300355-->
+END%%
+
+%%ANKI
+Basic
+What floor/ceiling identity does expression `(x + (1 << k) - 1) >> k` exploit?
+Back: $$\left\lceil \frac{x}{y} \right\rceil = \left\lfloor \frac{x + y - 1}{y} \right\rfloor$$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c17
+<!--ID: 1714184300359-->
+END%%
+
+%%ANKI
+Basic
+In two's-complement, how do we use `>>` to perform integer division of `x > 0` by $2^k$?
+Back: `x >> k`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c17
+<!--ID: 1714184300362-->
+END%%
+
+%%ANKI
+Basic
+In two's-complement, how do we use `>>` to perform integer division of `x < 0` by $2^k$?
+Back: `(x + (1 << k) - 1) >> k`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c17
+<!--ID: 1714184300364-->
+END%%
+
 ## Bibliography
 
 * Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.

notes/programming/assertions.md

@@ -0,0 +1,16 @@
+---
+title: Assertions
+TARGET DECK: Obsidian::STEM
+FILE TAGS: programming::assertions
+tags:
+  - assertions
+  - programming
+---
+
+## Overview
+
+TODO
+
+## Bibliography
+
+* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

notes/programming/equiv-trans.md

@@ -0,0 +1,547 @@
+---
+title: Equivalence Transformation
+TARGET DECK: Obsidian::STEM
+FILE TAGS: programming::equiv-trans
+tags:
+  - equiv-trans
+  - logic
+  - programming
+---
+
+## Overview
+
+**Equivalence-transformation** refers to a class of calculi for [[propositional|propositional logic]] derived from negation ($\neg$), conjunction ($\land$), disjunction ($\lor$), implication ($\Rightarrow$), and equality ($=$). Gries covers two in "The Science of Programming": a system of evaluation and a formal system. The system of evaluation mirrors how a computer processes instructions, at least in an abstract sense. The formal system serves as a theoretical framework for reasoning about propositions and their transformations without resorting to "lower-level" operations like substitution.
+
+%%ANKI
+Basic
+Who is the author of "The Science of Programming"?
+Back: David Gries
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861286-->
+END%%
+
+%%ANKI
+Basic
+What are constant propositions?
+Back: Propositions that contain only constants as operands.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707422675517-->
+END%%
+
+%%ANKI
+Cloze
+Gries replaces logical operator {$\Leftrightarrow$} in favor of {$=$}.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861295-->
+END%%
+
+%%ANKI
+Basic
+How does Lean define propositional equality?
+Back: Expressions `a` and `b` are propositionally equal iff `a = b` is true.
+Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
+Tags: lean
+<!--ID: 1706994861298-->
+END%%
+
+%%ANKI
+Basic
+How does Lean define `propext`?
+Back:
+```lean
+axiom propext {a b : Prop} : (a ↔ b) → (a = b)
+```
+Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
+Tags: lean
+<!--ID: 1706994861300-->
+END%%
+
+%%ANKI
+Basic
+What Lean theorem justifies Gries' choice of $=$ over $\Leftrightarrow$?
+Back: `propext`
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+Tags: lean
+<!--ID: 1706994861302-->
+END%%
+
+%%ANKI
+Basic
+Is $(b \land c)$ well-defined in $\{(b, T), (c, F)\}$?
+Back: Yes.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861318-->
+END%%
+
+%%ANKI
+Basic
+Is $(b \lor d)$ well-defined in $\{(b, T), (c, F)\}$?
+Back: No.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861320-->
+END%%
+
+%%ANKI
+Basic
+What proposition represents states $\{(b, T)\}$ and $\{(c, F)\}$?
+Back: $b \lor \neg c$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861337-->
+END%%
+
+%%ANKI
+Basic
+What set of states does $a \land b$ represent?
+Back: The set containing just state $\{(a, T), (b, T)\}$.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861339-->
+END%%
+
+%%ANKI
+Basic
+What is sloppy about phrase "the states in $b \lor \neg c$"?
+Back: $b \lor \neg c$ is not a set but a representation of a set (of states).
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861341-->
+END%%
+
+%%ANKI
+Basic
+What is the weakest proposition?
+Back: $T$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861348-->
+END%%
+
+%%ANKI
+Basic
+What set of states does $T$ represent?
+Back: The set of all states.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861350-->
+END%%
+
+%%ANKI
+Basic
+What is the strongest proposition?
+Back: $F$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861352-->
+END%%
+
+%%ANKI
+Basic
+What set of states does $F$ represent?
+Back: The set of no states.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861354-->
+END%%
+
+%%ANKI
+Basic
+What does a proposition *represent*?
+Back: The set of states in which it is true.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861335-->
+END%%
+
+%%ANKI
+Basic
+When is $p$ stronger than $q$?
+Back: When $p \Rightarrow q$.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861343-->
+END%%
+
+%%ANKI
+Basic
+When is $p$ weaker than $q$?
+Back: When $q \Rightarrow p$.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861346-->
+END%%
+
+%%ANKI
+Basic
+A proposition is well-defined with respect to what?
+Back: A state to evaluate against.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861316-->
+END%%
+
+%%ANKI
+Basic
+Why is $b \land c$ stronger than $b \lor c$?
+Back: The former represents a subset of the states the latter represents.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861356-->
+END%%
+
+%%ANKI
+Basic
+What is a state?
+Back: A function mapping identifiers to values.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861314-->
+END%%
+
+%%ANKI
+Basic
+What are the two calculi Gries describes equivalence-transformation with?
+Back: A formal system and a system of evaluation.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673342-->
+END%%
+
+## Equivalence Schemas
+
+A proposition is said to be a **tautology** if it evaluates to $T$ in every state it is well-defined in. We say propositions $E1$ and $E2$ are **equivalent** if $E1 = E2$ is a tautology. In this case, we say $E1 = E2$ is an **equivalence**.
+
+%%ANKI
+Basic
+What does it mean for a proposition to be a tautology?
+Back: That the proposition is true in every state it is well-defined in.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861323-->
+END%%
+
+%%ANKI
+Basic
+How is tautology $e$ written equivalently with a quantifier?
+Back: For free identifiers $i_1, \ldots, i_n$ in $e$, as $\forall (i_1, \ldots, i_n), e$.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707937867032-->
+END%%
+
+%%ANKI
+Basic
+The term "equivalent" refers to a comparison between what two objects?
+Back: Expressions.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673345-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for two propositions to be equivalent?
+Back: Given propositions $E1$ and $E2$, it means $E1 = E2$ is a tautology.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673347-->
+END%%
+
+%%ANKI
+Basic
+What is an equivalence?
+Back: Given propositions $E1$ and $E2$, tautology $E1 = E2$.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673348-->
+END%%
+
+* Commutative Laws
+	* $(E1 \land E2) = (E2 \land E1)$
+	* $(E1 \lor E2) = (E2 \lor E1)$
+	* $(E1 = E2) = (E2 = E1)$
+
+%%ANKI
+Basic
+Which of the basic logical operators do the commutative laws apply to?
+Back: $\land$, $\lor$, and $=$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673350-->
+END%%
+
+%%ANKI
+Basic
+What do the commutative laws allow us to do?
+Back: Reorder operands.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673351-->
+END%%
+
+%%ANKI
+Basic
+What is the commutative law of e.g. $\land$?
+Back: $E1 \land E2 = E2 \land E1$
+<!--ID: 1707251673353-->
+END%%
+
+* Associative Laws
+	* $E1 \land (E2 \land E3) = (E1 \land E2) \land E3$
+	* $E1 \lor (E2 \lor E3) = (E1 \lor E2) \lor E3$
+
+%%ANKI
+Basic
+Which of the basic logical operators do the associative laws apply to?
+Back: $\land$ and $\lor$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673354-->
+END%%
+
+%%ANKI
+Basic
+What do the associative laws allow us to do?
+Back: Remove parentheses.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673355-->
+END%%
+
+%%ANKI
+Basic
+What is the associative law of e.g. $\land$?
+Back: $E1 \land (E2 \land E3) = (E1 \land E2) \land E3$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673357-->
+END%%
+
+* Distributive Laws
+	* $E1 \lor (E2 \land E3) = (E1 \lor E2) \land (E1 \lor E3)$
+	* $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$
+
+%%ANKI
+Basic
+Which of the basic logical operators do the distributive laws apply to?
+Back: $\land$ and $\lor$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673358-->
+END%%
+
+%%ANKI
+Basic
+What do the distributive laws allow us to do?
+Back: "Factor" propositions.
+Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673360-->
+END%%
+
+%%ANKI
+Basic
+What is the distributive law of e.g. $\land$ over $\lor$?
+Back: $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673361-->
+END%%
+
+* De Morgan's Laws
+	* $\neg (E1 \land E2) = \neg E1 \lor \neg E2$
+	* $\neg (E1 \lor E2) = \neg E1 \land \neg E2$
+
+%%ANKI
+Basic
+Which of the basic logical operators do De Morgan's Laws apply to?
+Back: $\neg$, $\land$, and $\lor$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673363-->
+END%%
+
+%%ANKI
+Basic
+What is De Morgan's Law of e.g. $\land$?
+Back: $\neg (E1 \land E2) = \neg E1 \lor \neg E2$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673364-->
+END%%
+
+* Law of Negation
+	* $\neg (\neg E1) = E1$
+
+%%ANKI
+Basic
+What does the Law of Negation say?
+Back: $\neg (\neg E1) = E1$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673365-->
+END%%
+
+* Law of the Excluded Middle
+	* $E1 \lor \neg E1 = T$
+
+%%ANKI
+Basic
+Which of the basic logical operators does the Law of the Excluded Middle apply to?
+Back: $\lor$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673367-->
+END%%
+
+%%ANKI
+Basic
+What does the Law of the Excluded Middle say?
+Back: $E1 \lor \neg E1 = T$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673368-->
+END%%
+
+%%ANKI
+Basic
+Which equivalence schema is "refuted" by sentence, "This sentence is false."
+Back: Law of the Excluded Middle
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251779153-->
+END%%
+
+* Law of Contradiction
+	* $E1 \land \neg E1 = F$
+
+%%ANKI
+Basic
+Which of the basic logical operators does the Law of Contradiction apply to?
+Back: $\land$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673370-->
+END%%
+
+%%ANKI
+Basic
+What does the Law of Contradiction say?
+Back: $E1 \land \neg E1 = F$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673371-->
+END%%
+
+%%ANKI
+Cloze
+The Law of {1:the Excluded Middle} is to {2:$\lor$} whereas the Law of {2:Contradiction} is to {1:$\land$}.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707251673373-->
+END%%
+
+Gries lists other "Laws" but they don't seem as important to note here.
+
+%%ANKI
+Basic
+How is $\Rightarrow$ written in terms of other logical operators?
+Back: $p \Rightarrow q$ is equivalent to $\neg p \lor q$.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861358-->
+END%%
+
+%%ANKI
+Basic
+How is $\Leftrightarrow$/$=$ written in terms of other logical operators?
+Back: $p \Leftrightarrow q$ is equivalent to $(p \Rightarrow q) \land (q \Rightarrow p)$.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861360-->
+END%%
+
+%%ANKI
+Basic
+What distinguishes an equality from an equivalence?
+Back: An equivalence is an equality that is also a tautology.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707316178709-->
+END%%
+
+## Equivalence Rules
+
+* Rule of Substitution
+	* Let $P(r)$ be a predicate and $E1 = E2$ be an equivalence. Then $P(E1) = P(E2)$ is an equivalence.
+* Rule of Transitivity
+	* Let $E1 = E2$ and $E2 = E3$ be equivalences. Then $E1 = E3$ is an equivalence.
+
+%%ANKI
+Basic
+What two inference rules make up the equivalence-transformation formal system?
+Back: Substitution and transitivity.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707253246450-->
+END%%
+
+%%ANKI
+Basic
+Which of the two inference rules that make up the equivalence-transformation formal system is redundant?
+Back: Transitivity.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707432641598-->
+END%%
+
+%%ANKI
+Basic
+What does the rule of substitution say in the system of evaluation?
+Back: Let $P(r)$ be a predicate and $E1 = E2$ be an equivalence. Then $P(E1) = P(E2)$ is an equivalence.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707253246452-->
+END%%
+
+%%ANKI
+Basic
+How is the rule of substitution written as an inference rule (in standard form)?
+Back:
+$$
+\begin{matrix}
+E1 = E2 \\
+\hline P(E1) = P(E2)
+\end{matrix}
+$$
+<!--ID: 1707253246454-->
+END%%
+
+%%ANKI
+Basic
+What does the rule of transitivity state in the system of evaluation?
+Back: Let $E1 = E2$ and $E2 = E3$. Then $E1 = E3$.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707253246455-->
+END%%
+
+%%ANKI
+Basic
+How is the rule of transitivity written as an inference rule (in standard form)?
+Back:
+$$
+\begin{matrix}
+E1 = E2, E2 = E3 \\
+\hline E1 = E3
+\end{matrix}
+$$
+<!--ID: 1707253246457-->
+END%%
+
+%%ANKI
+Cloze
+The system of evaluation has {equivalences} whereas the formal system has {theorems}.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707253246458-->
+END%%
+
+%%ANKI
+Basic
+What is a "theorem" in the equivalence-transformation formal system?
+Back: An equivalence derived from the axioms and inference rules.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707316178712-->
+END%%
+
+%%ANKI
+Basic
+How is e.g. the Law of Implication proven in the system of evaluation?
+Back: With truth tables
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707316178714-->
+END%%
+
+%%ANKI
+Basic
+How is e.g. the Law of Implication proven in the formal system?
+Back: It isn't. It is an axiom.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707316178715-->
+END%%
+
+%%ANKI
+Cloze
+The system of evaluation and formal system are connected by the following biconditional: {$e$ is a tautology} iff {$e = T$ is a theorem}.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707316178717-->
+END%%
+
+%%ANKI
+Cloze
+The {1:system of evaluation} is to {2:"$e$ is a tautology"} whereas the {2:formal system} is to {1:"$e = T$ is a theorem"}.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707316276203-->
+END%%
+
+## Bibliography
+
+* Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
+* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

notes/programming/index.md

@@ -0,0 +1,5 @@
+---
+title: Programming
+tags:
+  - programming
+---

notes/set/trees.md

@@ -960,7 +960,7 @@ END%%
 %%ANKI
 Basic
 What does it mean for a $k$-ary tree to be complete?
-Back: The last level is not filled but all leaves have the same depth and are leftmost arranged.
+Back: All levels, except maybe the last, are filled. All leaves have the same depth and are leftmost arranged.
 Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 <!--ID: 1714080353480-->
 END%%
@@ -989,6 +989,86 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
 <!--ID: 1714082676022-->
 END%%
 
+%%ANKI
+Basic
+How many internal nodes are in a complete $k$-ary tree of $n$ nodes?
+Back: $\lceil (n - 1) / k \rceil$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714349367630-->
+END%%
+
+%%ANKI
+Basic
+What value of $k$ is used in the following description of a complete $k$-ary tree?
+$$\begin{array}{c|c|c}
+n & \text{external} & \text{internal} \\
+\hline
+1 & 1 & 0 \\
+2 & 1 & 1 \\
+3 & 2 & 1 \\
+4 & 3 & 1 \\
+5 & 4 & 1 \\
+6 & 4 & 2 \\
+7 & 5 & 2 \\
+8 & 6 & 2
+\end{array}$$
+Back: $4$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714349367637-->
+END%%
+
+%%ANKI
+Basic
+What value of $k$ is used in the following description of a complete $k$-ary tree?
+$$\begin{array}{c|c|c}
+n & \text{external} & \text{internal} \\
+\hline
+1 & 1 & 0 \\
+2 & 1 & 1 \\
+3 & 2 & 1 \\
+4 & 2 & 2 \\
+5 & 3 & 2 \\
+6 & 3 & 3 \\
+7 & 4 & 3 \\
+8 & 4 & 4
+\end{array}$$
+Back: $2$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714349367640-->
+END%%
+
+%%ANKI
+Basic
+When does the number of external nodes increment in a growing $k$-ary tree?
+Back: When the next node added already has a sibling.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714349367644-->
+END%%
+
+%%ANKI
+Basic
+When does the number of external nodes remain static in a growing $k$-ary tree?
+Back: When the next node added has no sibling.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714349367647-->
+END%%
+
+%%ANKI
+Basic
+When does the number of internal nodes increment in a growing $k$-ary tree?
+Back: When the next node added has no sibling.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714349367651-->
+END%%
+
+%%ANKI
+Basic
+When does the number of internal nodes remain static in a growing $k$-ary tree?
+Back: When the next node added already has a sibling.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714349367655-->
+END%%
+
 #### Binary Trees
 
 A **binary tree** $T$ is a structure defined on a finite set of nodes that either
@@ -1165,7 +1245,7 @@ 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: 1714080353484-->
+<!--ID: 1714180419777-->
 END%%
 
 %%ANKI
@@ -1174,7 +1254,7 @@ 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: 1714080353488-->
+<!--ID: 1714180419781-->
 END%%
 
 %%ANKI
@@ -1183,7 +1263,7 @@ 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: 1714088438768-->
+<!--ID: 1714180419784-->
 END%%
 
 %%ANKI
@@ -1192,7 +1272,7 @@ 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: 1714080353491-->
+<!--ID: 1714180419787-->
 END%%
 
 %%ANKI
@@ -1201,7 +1281,7 @@ 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: 1714080353495-->
+<!--ID: 1714180419789-->
 END%%
 
 %%ANKI
@@ -1210,7 +1290,7 @@ 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: 1714088438773-->
+<!--ID: 1714180419793-->
 END%%
 
 %%ANKI
@@ -1219,7 +1299,7 @@ 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: 1714080353498-->
+<!--ID: 1714180419802-->
 END%%
 
 %%ANKI
@@ -1228,7 +1308,7 @@ 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: 1714080353502-->
+<!--ID: 1714180419809-->
 END%%
 
 %%ANKI
@@ -1237,7 +1317,7 @@ 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: 1714088438777-->
+<!--ID: 1714180419813-->
 END%%
 
 %%ANKI
@@ -1366,6 +1446,14 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
 <!--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%%
+
 ## 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).