Refine notes on complete/perfect trees.

2024-04-25 18:01:06 -06:00 · 2024-04-25 18:01:06 -06:00 · 9ee37c8b7d
parent 262683515f
commit 9ee37c8b7d
10 changed files with 678 additions and 260 deletions
--- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
+++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
@ -109,7 +109,12 @@
    "rooted-tree.png",
    "ordered-rooted-tree.png",
    "ordered-rooted-tree-cmp.png",
-    "ordered-binary-tree-cmp.png"
+    "ordered-binary-tree-cmp.png",
+    "complete-tree.png",
+    "nearly-complete-tree.png",
+    "non-nearly-complete-tree.png",
+    "perfect-tree.png",
+    "non-complete-tree.png"
  ],
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@ -221,10 +226,10 @@
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-    "c17/strings.md": "e4012ee9a18ad44370490e375ae57f09",
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@ -246,7 +251,7 @@
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@ -310,7 +315,7 @@
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    "git/refs.md": "954fc69004aa65b358ec5ce07c1435ce",
-    "set/trees.md": "4d081c258eb662e2147d1e462193e715",
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@ -362,8 +367,11 @@
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--- a/notes/_journal/2024-04/2024-04-24.md
+++ b/notes/_journal/2024-04/2024-04-24.md
@ -15,4 +15,5 @@ title: "2024-04-24"
 	* Explored how to limit the number of re-renders in the application.
 	* Played around with Zustand types to get TypeScript and Immer working together (WIP).
 * Read chapter 2 "Pastebin" of "Grokking the System Design Interview".
-* Read chapter 4 of HP-16C manual.
+* Read chapter 4 of HP-16C manual.
+* Add notes on iterating recurrence relations.
--- a/notes/_journal/2024-04/2024-04-25.md
+++ b/notes/_journal/2024-04/2024-04-25.md
@ -0,0 +1,15 @@
+---
+title: "2024-04-25"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] Sheet Music (10 min.)
+- [ ] Go (1 Life & Death Problem)
+- [ ] Korean (Read 1 Story)
+- [x] Interview Prep (1 Practice Problem)
+- [x] Log Work Hours (Max 3 hours)
+
+* Talk with Mike about yesterday's "Grokking the System Interview" and "Designing Data-Intensive Applications" reading.
+* Notes on nearly complete trees. Refine terminology to favor "perfect/complete" over "complete/nearly complete".
+* Read Heapsort section in "Introduction to Algorithms". Still need to implement and translate into notes though.
--- a/notes/algebra/sequences/index.md
+++ b/notes/algebra/sequences/index.md
@ -171,6 +171,12 @@ END%%

 ## Solving Recurrence Relations

+We use three different strategies for solving recurrences:
+
+* Telescoping
+* Iteration
+* Characteristic Polynomials
+
 %%ANKI
 Basic
 What is the recurrence relation for the Fibonacci sequence?
@ -245,15 +251,14 @@ END%%

 %%ANKI
 Basic
-Schematically show how telescoping can be used to solve $a_n = a_{n-1} + f(n)$.
+Schematically show how **telescoping** can be used to solve $a_n = a_{n-1} + f(n)$.
 Back: $$\begin{align*}
 a_1 - a_0 & = f(1) \\
 & \vdots \\
 a_n - a_{n-1} & = f(n) \\
 \hline
 a_n - a_0 & = \sum_{k=1}^n f(k)
-\end{align*}$$
-Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+\end{align*}$$Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
 <!--ID: 1713810280088-->
 END%%

@ -273,6 +278,71 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
 <!--ID: 1713810280094-->
 END%%

+%%ANKI
+Basic
+What does it mean to solve a recurrence relation using iteration?
+Back: Repeatedly expand terms, starting at the initial conditions, to discover a pattern.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1713998412589-->
+END%%
+
+%%ANKI
+Basic
+What is the result of "iterating" the following recursive definition twice? $$a_n = 3a_{n-1} + 2$$
+Back: $$\begin{align*}
+a_1 & = 3(a_0) + 2 \\
+a_2 & = 3(3(a_0) + 2) + 2
+\end{align*}$$
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1713998412595-->
+END%%
+
+%%ANKI
+Basic
+Schematically show how **iteration** can be used to solve $a_n = a_{n-1} + f(n)$.
+Back: $$\begin{align*}
+a_1 & = a_0 + f(1) \\
+& \vdots \\
+a_n & = (\cdots((a_0 + f(1)) + f(2)) + \cdots) + f(n) \\
+\hline
+a_n & = a_0 + \sum_{k=1}^n f(k)
+\end{align*}$$
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1713998412597-->
+END%%
+
+%%ANKI
+Basic
+How fast does the number of terms grow when iterating $a_n = 3a_{n-1} + 2$?
+Back: Linearly.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1713998412600-->
+END%%
+
+%%ANKI
+Basic
+How fast does the number of terms grow when iterating $a_n = 2a_{n-1} + 3a_{n-2}$?
+Back: Exponentially (the number of terms double each iteration).
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1713998412603-->
+END%%
+
+%%ANKI
+Basic
+Why should you avoid using iteration to solve $a_n = 2a_{n-1} + 3a_{n-2}$?
+Back: The number of terms grows unwieldy very quickly.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1713998412607-->
+END%%
+
+%%ANKI
+Basic
+When solving recurrences, is telescoping or iteration a more general technique?
+Back: Iteration.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1713998536738-->
+END%%
+
 ## Bibliography

 * Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
--- a/notes/algorithms/heaps.md
+++ b/notes/algorithms/heaps.md
@ -0,0 +1,12 @@
+---
+title: Heaps
+TARGET DECK: Obsidian::STEM
+FILE TAGS: algorithm::data_structure
+tags:
+  - algorithm
+  - data_structure
+---
+
+## Overview
+
+TODO
--- a/notes/c17/strings.md
+++ b/notes/c17/strings.md
@ -36,7 +36,7 @@ END%%

 %%ANKI
 Basic
-Text is more platform-independent than binary because it is unaffected by what two properties?
+Text is more platform-independent than e.g. integer encodings because it is unaffected by what two properties?
 Back: Word size and byte ordering.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1707758281270-->
--- a/notes/set/images/complete-tree.png
+++ b/notes/set/images/complete-tree.png
--- a/notes/set/images/non-complete-tree.png
+++ b/notes/set/images/non-complete-tree.png
--- a/notes/set/images/perfect-tree.png
+++ b/notes/set/images/perfect-tree.png
--- a/notes/set/trees.md
+++ b/notes/set/trees.md
@ -1,10 +1,11 @@
 ---
 title: Trees
 TARGET DECK: Obsidian::STEM
-FILE TAGS: set::graph
+FILE TAGS: set::tree
 tags:
  - graph
  - set
+  - tree
 ---

 ## Overview
@ -153,6 +154,22 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
 <!--ID: 1711136844955-->
 END%%

+%%ANKI
+Basic
+How many levels exist in a rooted tree of height $h$?
+Back: $h + 1$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713118128242-->
+END%%
+
+%%ANKI
+Basic
+What is the height of a rooted tree with $k$ levels?
+Back: $k - 1$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713118128244-->
+END%%
+
 %%ANKI
 Basic
 Which free trees are not considered rooted trees?
@ -644,16 +661,16 @@ END%%

 %%ANKI
 Basic
-The following two trees are equivalent when considered as what kind of trees?
+The following two trees are equivalent when considered as what (most specific) kind of trees?
 ![[ordered-rooted-tree-cmp.png]]
-Back: Rooted/free trees.
+Back: Rooted trees.
 Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 <!--ID: 1712407152755-->
 END%%

 %%ANKI
 Basic
-The following two trees are different when considered as what kind of trees?
+The following two trees are different when considered as what (most general) kind of trees?
 ![[ordered-rooted-tree-cmp.png]]
 Back: Ordered trees.
 Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
@ -678,6 +695,15 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
 <!--ID: 1712409466670-->
 END%%

+%%ANKI
+Basic
+Considered as positional trees, are the following trees the same?
+![[ordered-binary-tree-cmp.png]]
+Back: No.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714089436122-->
+END%%
+
 %%ANKI
 Basic
 Considered as binary trees, are the following trees the same?
@ -687,13 +713,541 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
 <!--ID: 1712409466676-->
 END%%

-### Binary Trees
+%%ANKI
+Basic
+Why are these two binary trees not the same?
+![[ordered-binary-tree-cmp.png]]
+Back: `5` is a left child in the first tree but a right child in the second.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1712409466682-->
+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 $k$-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.
+
+%%ANKI
+Basic
+Why aren't terms "complete/perfect" and "nearly complete/complete" quite synonymous?
+Back: In the former, "perfect" trees are a subset of "complete" trees.
+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: 1714088438740-->
+END%%
+
+%%ANKI
+Basic
+What distinguishes a positional tree from a $k$-ary tree?
+Back: A $k$-ary tree cannot have child with label $> k$.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713118128216-->
+END%%
+
+%%ANKI
+Basic
+Is a $k$-ary tree a positional tree?
+Back: Yes.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714089436130-->
+END%%
+
+%%ANKI
+Basic
+Is a positional tree a $k$-ary tree?
+Back: Not necessarily.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714089436134-->
+END%%
+
+%%ANKI
+Basic
+What distinguishes positional trees from ordered trees?
+Back: Children of the former are labeled with a distinct positive integer.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713118128219-->
+END%%
+
+%%ANKI
+Basic
+Is the notion of absent children a concept in ordered trees?
+Back: No.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714088438749-->
+END%%
+
+%%ANKI
+Basic
+Is the notion of absent children a concept in positional trees?
+Back: Yes.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714088438754-->
+END%%
+
+%%ANKI
+Basic
+Is the notion of absent children a concept in $k$-ary trees?
+Back: Yes.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714088438759-->
+END%%
+
+%%ANKI
+Basic
+What is a positional tree?
+Back: A rooted tree in which each child is labeled with a distinct positive integer.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713118128220-->
+END%%
+
+%%ANKI
+Basic
+What is a $k$-ary tree?
+Back: A positional tree with labels greater than $k$ missing.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713118128223-->
+END%%
+
+%%ANKI
+Basic
+Which of positional trees or $k$-ary trees are more general? 
+Back: The positional tree.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713118128225-->
+END%%
+
+%%ANKI
+Basic
+Which of positional trees or ordered trees are more general? 
+Back: N/A.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714088438763-->
+END%%
+
+%%ANKI
+Is the concept of fullness related to positional trees or $k$-ary trees?
+Back: $k$-ary trees.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+END%%
+
+%%ANKI
+Basic
+Is the concept of perfectness related to positional trees or $k$-ary trees?
+Back: $k$-ary trees.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713118128229-->
+END%%
+
+%%ANKI
+Basic
+Is the concept of completeness related to positional trees or $k$-ary trees?
+Back: $k$-ary trees.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714088723844-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for a $k$-ary tree to be full?
+Back: Each node has $0$ or $k$ children.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713118128231-->
+END%%
+
+%%ANKI
+Basic
+What degrees are permitted in a full $k$-ary tree?
+Back: $0$ or $k$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713118128233-->
+END%%
+
+%%ANKI
+Basic
+What degrees are permitted in a perfect $k$-ary tree?
+Back: $0$ or $k$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713118128234-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for a $k$-ary tree to be perfect?
+Back: All leaves have the same depth and all internal nodes have degree $k$.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713118128236-->
+END%%
+
+%%ANKI
+Basic
+What is the degree of an internal node in a perfect $k$-ary tree?
+Back: $k$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713118128239-->
+END%%
+
+%%ANKI
+Basic
+What is the degree of an external node in a perfect $k$-ary tree?
+Back: $0$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713118128241-->
+END%%
+
+%%ANKI
+Basic
+What recursive definition describes the number of nodes in each level of a perfect $k$-ary tree?
+Back: $a_n = k \cdot a_{n-1}$ with $a_0 = 1$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+Tags: algebra::sequence
+<!--ID: 1713118128248-->
+END%%
+
+%%ANKI
+Basic
+How many nodes are in a perfect $k$-ary tree of height $h$?
+Back: $$\frac{1 - k^{h+1}}{1 - k}$$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+Tags: algebra::sequence
+<!--ID: 1713118128249-->
+END%%
+
+%%ANKI
+Basic
+How many internal nodes are in a perfect $k$-ary tree of height $h$?
+Back: $$\frac{1 - k^h}{1 - k}$$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+Tags: algebra::sequence
+<!--ID: 1714080353459-->
+END%%
+
+%%ANKI
+Basic
+How many external nodes are in a perfect $k$-ary tree of height $h$?
+Back: $k^h$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+Tags: algebra::sequence
+<!--ID: 1714080353455-->
+END%%
+
+%%ANKI
+Basic
+How many nodes are on level $d$ of a perfect $k$-ary tree?
+Back: $k^d$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+Tags: algebra::sequence
+<!--ID: 1714080353462-->
+END%%
+
+%%ANKI
+Basic
+What kind of sequence describes the number of nodes in a perfect $k$-ary tree?
+Back: A geometric sequence.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+Tags: algebra::sequence
+<!--ID: 1713118128251-->
+END%%
+
+%%ANKI
+Basic
+What is the common ratio of the geometric sequence used to count nodes of a perfect $k$-ary tree?
+Back: $k$
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+Tags: algebra::sequence
+<!--ID: 1713118128253-->
+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.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714080353480-->
+END%%
+
+%%ANKI
+Basic
+How is the minimum number of nodes in a complete $k$-ary tree of height $h$ calculated in terms of perfect $k$-ary trees?
+Back: As "the number of nodes in a perfect $k$-ary tree of height $h - 1$" plus $1$.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714082676018-->
+END%%
+
+%%ANKI
+Basic
+What is the maximum number of nodes in a complete 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).
+<!--ID: 1714082676014-->
+END%%
+
+%%ANKI
+Basic
+How is the maximum number of nodes in a complete $k$-ary tree of height $h$ calculated in terms of perfect $k$-ary trees?
+Back: As "the number of nodes in a perfect $k$-ary tree of height $h$".
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714082676022-->
+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: 1714080353484-->
+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: 1714080353488-->
+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: 1714088438768-->
+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: 1714080353491-->
+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: 1714080353495-->
+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: 1714088438773-->
+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: 1714080353498-->
+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: 1714080353502-->
+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: 1714088438777-->
+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?
@ -812,249 +1366,7 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
 <!--ID: 1712409466653-->
 END%%

-%%ANKI
-Is a binary tree a $k$-ary tree?
-Back: Yes.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-END%%
-
-ANKI%%
-Is a $k$-ary tree a positional tree?
-Back: Yes.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-END%%
-
-ANK%%
-Is a positional tree a $k$-ary tree?
-Back: Not necessarily.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-END%%
-
-%%ANKI
-Basic
-Why are these two binary trees not the same?
-![[ordered-binary-tree-cmp.png]]
-Back: `5` is a left child in the first tree but a right child in the second.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1712409466682-->
-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 $k$-ary tree is **full** if every node has degree $0$ or $k$. A $k$-ary tree is **complete** if all leaves have the same depth and all internal nodes have degree $k$.
-
-%%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
-What does it mean for a binary tree to be complete?
-Back: Each leaf has the same depth and all internal nodes have degree $2$.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-END%%
-
-%%ANKI
-What degrees are permitted in a full binary tree?
-Back: $0$ or $2$
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-END%%
-
-%%ANKI
-What degrees are permitted in a complete binary tree?
-Back: $0$ or $2$
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-END%%
-
-%%ANKI
-What category of rooted tree does a binary tree fall under?
-Back: A positional tree or $k$-ary tree.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-END%%
-
-%%ANKI
-Basic
-What distinguishes a positional tree from a $k$-ary tree?
-Back: A $k$-ary tree is a positional tree in which each node has at most $k$ children.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1713118128216-->
-END%%
-
-%%ANKI
-Basic
-What distinguishes positional trees from ordered trees?
-Back: The same children in different positions is considered distinct in the former.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1713118128219-->
-END%%
-
-%%ANKI
-Basic
-What is a positional tree?
-Back: A rooted tree in which each child is labeled with a specific positive integer.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1713118128220-->
-END%%
-
-%%ANKI
-Basic
-What is a $k$-ary tree?
-Back: A positional tree in which each node has $k$ labels with a potential child.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1713118128223-->
-END%%
-
-%%ANKI
-Basic
-Which of positional trees or $k$-ary trees are more general? 
-Back: The positional tree.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1713118128225-->
-END%%
-
-%%ANKI
-Basic
-Which of positional trees or $k$-ary trees are more general? 
-Back: The positional tree.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-END%%
-
-%%ANKI
-Basic
-Is a binary tree a positional tree?
-Back: Yes.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1713118128227-->
-END%%
-
-%%ANKI
-Is the concept of fullness related to positional trees or $k$-ary trees?
-Back: $k$-ary trees.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-END%%
-
-%%ANKI
-Basic
-Is the concept of completeness related to positional trees or $k$-ary trees?
-Back: $k$-ary trees.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1713118128229-->
-END%%
-
-%%ANKI
-Basic
-What does it mean for a $k$-ary tree to be full?
-Back: Each node has $0$ or $k$ children.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1713118128231-->
-END%%
-
-%%ANKI
-Basic
-What degrees are permitted in a full $k$-ary tree?
-Back: $0$ or $k$
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1713118128233-->
-END%%
-
-%%ANKI
-Basic
-What degrees are permitted in a complete $k$-ary tree?
-Back: $0$ or $k$
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1713118128234-->
-END%%
-
-%%ANKI
-Basic
-What does it mean for a $k$-ary tree to be complete?
-Back: All leaves have the same depth and all internal nodes have degree $k$.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1713118128236-->
-END%%
-
-%%ANKI
-Basic
-What is the degree of an internal node in a complete $k$-ary tree'?
-Back: $k$
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1713118128239-->
-END%%
-
-%%ANKI
-Basic
-What is the degree of an external node in a complete $k$-ary tree'?
-Back: $0$
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1713118128241-->
-END%%
-
-%%ANKI
-Basic
-How many levels exist in a rooted tree of height $h$?
-Back: $h + 1$
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1713118128242-->
-END%%
-
-%%ANKI
-Basic
-What is the height of a rooted tree with $k$ levels?
-Back: $k - 1$
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1713118128244-->
-END%%
-
-%%ANKI
-Basic
-What recursive definition describes the number of nodes in each level of a complete $k$-ary tree?
-Back: $a_n = k \cdot a_{n-1}$ with $a_0 = 1$
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-Tags: algebra::sequence
-<!--ID: 1713118128248-->
-END%%
-
-%%ANKI
-Basic
-What closed formula details the number of nodes in a complete $k$-ary tree of height $h$?
-Back: $$\frac{1 - k^h}{1 - k}$$
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-Tags: algebra::sequence
-<!--ID: 1713118128249-->
-END%%
-
-%%ANKI
-Basic
-What kind of sequence describes the number of nodes in a complete $k$-ary tree?
-Back: A geometric sequence.
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-Tags: algebra::sequence
-<!--ID: 1713118128251-->
-END%%
-
-%%ANKI
-Basic
-What is the common ratio in the geometric sequence counting nodes of a complete $k$-ary tree?
-Back: $k$
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-Tags: algebra::sequence
-<!--ID: 1713118128253-->
-END%%
-
-%%ANKI
-Basic
-How many nodes are in a complete binary tree of height $h$?
-Back: $$\frac{1 - 2^h}{1 - 2} = 2^h - 1$$
-Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-Tags: algebra::sequence
-<!--ID: 1713118128255-->
-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).