Add positional tree notes.

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

@@ -148,7 +148,7 @@
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     "_journal/2024-01-31.md": "7c7fbfccabc316f9e676826bf8dfe970",
-    "logic/equiv-trans.md": "1198fbea260527e7dfd17b72955f5bee",
+    "logic/equiv-trans.md": "4de3b5412f29b080a5f9f3629db38389",
     "_journal/2024-02-07.md": "8d81cd56a3b33883a7706d32e77b5889",
     "algorithms/loop-invariants.md": "cbefc346842c21a6cce5c5edce451eb2",
     "algorithms/loop-invariant.md": "3b390e720f3b2a98e611b49a0bb1f5a9",
@@ -310,7 +310,7 @@
     "x86-64/declarations.md": "75bc7857cf2207a40cd7f0ee056af2f2",
     "x86-64/instructions.md": "240b4ceddf174f48207ba6bed4d25246",
     "git/refs.md": "954fc69004aa65b358ec5ce07c1435ce",
-    "set/trees.md": "fd092c77c26d0b99b3131da6dc043edd",
+    "set/trees.md": "0e2d146f7e952aa4575e4daa82301ecd",
     "_journal/2024-03-24.md": "1974cdb9fc42c3a8bebb8ac76d4b1fd6",
     "_journal/2024-03/2024-03-23.md": "ad4e92cc2bf37f174a0758a0753bf69b",
     "_journal/2024-03/2024-03-22.md": "a509066c9cd2df692549e89f241d7bd9",
@@ -334,13 +334,16 @@
     "_journal/2024-04-06.md": "1e3e0aa30b92987e6d53d5a366963b95",
     "_journal/2024-04/2024-04-05.md": "c829cead37580ed6ece0f98497d71147",
     "_journal/2024-04-11.md": "23b873dd315e716ad9ee5f107f9cfefc",
-    "_journal/2024-04/2024-04-10.md": "8ada51600e1bdafd37a5552ae67281e6",
+    "_journal/2024-04/2024-04-10.md": "e91fe6eb3d2ef5af5a293e4f92022faa",
     "_journal/2024-04/2024-04-09.md": "20eeafed43b6bc48786133c3523e1029",
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     "_journal/2024-04-12.md": "18955288bcdacf24c3d9d4b2a5b2e6d7",
-    "_journal/2024-04/2024-04-11.md": "9c248442a03080cf2942436a617edfef"
+    "_journal/2024-04/2024-04-11.md": "5baeb12bc5cc5492af766def8f849fb4",
+    "_journal/2024-04-14.md": "b51ad215e566bd96617d2817484232eb",
+    "_journal/2024-04/2024-04-13.md": "92db6bcb162a85fb0a4afd51999b6318",
+    "_journal/2024-04/2024-04-12.md": "1ec32792d88d470c2d517e5caf64fd6f"
   },
   "fields_dict": {
     "Basic": [

notes/_journal/2024-04-14.md

@@ -0,0 +1,11 @@
+---
+title: "2024-04-14"
+---
+
+- [ ] Anki Flashcards
+- [ ] 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)

notes/_journal/2024-04/2024-04-13.md

@@ -9,3 +9,5 @@ title: "2024-04-13"
 - [ ] Korean (Read 1 Story)
 - [ ] Interview Prep (1 Practice Problem)
 - [x] Log Work Hours (Max 3 hours)
+
+* Play tested the hide-and-seek application

notes/set/trees.md

@@ -121,6 +121,8 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
 <!--ID: 1711136844943-->
 END%%
 
+## Rooted Trees
+
 A **rooted tree** is a free tree in which one vertex is distinguished/blessed as the **root**. We call vertices of rooted trees **nodes**.
 
 Let $T$ be a rooted tree with root $r$. Any node $y$ on the simple path from $r$ to node $x$ is an **ancestor** of $x$. Likewise, $x$ is a **descendant** of $y$. If the last edge on the path from $r$ to $x$ is $\{y, x\}$, $y$ is the **parent** of $x$ and $x$ is a **child** of $y$. Nodes with the same parent are called **siblings**.
@@ -596,6 +598,8 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
 <!--ID: 1711136845240-->
 END%%
 
+### Ordered Trees
+
 An **ordered tree** is a rooted tree in which the children of each node are ordered.
 
 %%ANKI
@@ -656,6 +660,35 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
 <!--ID: 1712407152763-->
 END%%
 
+%%ANKI
+Basic
+Considered as rooted trees, are the following trees the same?
+![[ordered-binary-tree-cmp.png]]
+Back: Yes.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1712409466660-->
+END%%
+
+%%ANKI
+Basic
+Considered as ordered trees, are the following trees the same?
+![[ordered-binary-tree-cmp.png]]
+Back: Yes.
+Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1712409466670-->
+END%%
+
+%%ANKI
+Basic
+Considered as binary 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: 1712409466676-->
+END%%
+
+### Binary Trees
+
 A **binary tree** $T$ is a structure defined on a finite set of nodes that either
 
 * contains no nodes, or
@@ -780,30 +813,21 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
 END%%
 
 %%ANKI
-Basic
+Is a binary tree a $k$-ary tree?
-Considered as rooted trees, are the following trees the same?
-![[ordered-binary-tree-cmp.png]]
 Back: Yes.
 Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1712409466660-->
 END%%
 
-%%ANKI
+ANKI%%
-Basic
+Is a $k$-ary tree a positional tree?
-Considered as ordered trees, are the following trees the same?
-![[ordered-binary-tree-cmp.png]]
 Back: Yes.
 Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
-<!--ID: 1712409466670-->
 END%%
 
-%%ANKI
+ANK%%
-Basic
+Is a positional tree a $k$-ary tree?
-Considered as binary trees, are the following trees the same?
+Back: Not necessarily.
-![[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: 1712409466676-->
 END%%
 
 %%ANKI
@@ -815,6 +839,231 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
 <!--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 at most $k$ children.
+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 levelof 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: 1713118128246-->
+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
 
 * Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).