Notes on binary search trees.

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

@@ -146,7 +146,15 @@
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     "function-kernel.png",
     "triple-table-repr.png",
-    "binary-search-tree.png"
+    "binary-search-tree.png",
+    "bst-right-child.png",
+    "bst-right-child-after.png",
+    "bst-left-child.png",
+    "bst-left-child-after.png",
+    "bst-right-succ.png",
+    "bst-right-succ-after.png",
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notes/_journal/2024-08-03.md

@@ -2,8 +2,10 @@
 title: "2024-08-03"
 ---
 
-- [ ] Anki Flashcards
+- [x] Anki Flashcards
 - [x] KoL
 - [ ] OGS
 - [ ] Sheet Music (10 min.)
-- [ ] Korean (Read 1 Story)
+- [ ] Korean (Read 1 Story)
+
+* Additional notes on binary search trees.

notes/algebra/radices.md

@@ -277,7 +277,7 @@ END%%
 %%ANKI
 Basic
 Which hexadecimal digits are encoded in binary with a leading `1` bit?
-Back: `8` through `F`
+Back: `8` through `F`.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 Tags: binary::hex
 <!--ID: 1708631918825-->
@@ -286,7 +286,7 @@ END%%
 %%ANKI
 Basic
 Which hexadecimal digits are encoded in binary with a leading `0` bit?
-Back: `0` through `7`
+Back: `0` through `7`.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 Tags: binary::hex
 <!--ID: 1708631918829-->

notes/data-structures/binary-search-tree.md

@@ -13,6 +13,8 @@ A binary search tree (BST) is a [[trees#Binary Trees|binary tree]] satisfying th
 
 > Let $x$ be a node in a binary search tree. If $y$ is a node in the left subtree of $x$, then $y.key \leq x.key$. If $y$ is a node in the right subtree of $x$, then $y.key \geq x.key$.
 
+## Traversals
+
 Consider an arbitrary node $x$ of some BST. Then:
 
 * An **inorder** traversal visits $x$'s left child, then $x$, then $x$'s right child.
@@ -190,6 +192,398 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
 <!--ID: 1722342235784-->
 END%%
 
+%%ANKI
+Basic
+In terms of the height $h$ of a BST, what is the runtime of search?
+Back: $O(h)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303021-->
+END%%
+
+%%ANKI
+Basic
+In terms of the height $h$ of a BST, what is the runtime for finding the minimum?
+Back: $O(h)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303023-->
+END%%
+
+%%ANKI
+Basic
+In terms of the height $h$ of a BST, what is the runtime for finding the maximum?
+Back: $O(h)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303025-->
+END%%
+
+### Successors
+
+The **successor** of a node in a binary search tree is the node whose value would appear immediately after in an in-order traversal.
+
+%%ANKI
+Basic
+How do we define the successor of a BST node in terms of in-order traversals?
+Back: As the node encountered immediately after in an in-order traversal.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623599-->
+END%%
+
+%%ANKI
+Basic
+Which node is the successor of $7$ in the following BST?
+![[binary-search-tree.png]]
+Back: $8$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623601-->
+END%%
+
+%%ANKI
+Basic
+Which node is the successor of $2$ in the following BST?
+![[binary-search-tree.png]]
+Back: $5$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623603-->
+END%%
+
+%%ANKI
+Basic
+Which node is the successor of $5$ in the following BST?
+![[binary-search-tree.png]]
+Back: Either $5$ or $6$.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623604-->
+END%%
+
+%%ANKI
+Basic
+Which node(s) in a BST have no successor?
+Back: The maximum.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623606-->
+END%%
+
+%%ANKI
+Basic
+What two cases are considered when finding the successor of a BST node?
+Back: If the node has a right subtree or not.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623607-->
+END%%
+
+%%ANKI
+Basic
+Suppose a BST node has a right subtree. What is its successor?
+Back: The minimum of the right subtree.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623608-->
+END%%
+
+%%ANKI
+Basic
+Suppose a BST node does not have a right subtree. What is its successor?
+Back: The first proper ancestor reached from the LHS.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623610-->
+END%%
+
+%%ANKI
+Cloze
+If a BST node has a right subtree, it's successor cannot have a {left} subtree.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303026-->
+END%%
+
+%%ANKI
+Basic
+Suppose a BST node has a right subtree. *Why* can't its successor have a left subtree?
+Back: Because then a node in that left subtree would be the actual successor.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303028-->
+END%%
+
+%%ANKI
+Basic
+Suppose a BST node has a right subtree. *Why* can't its successor have a right subtree?
+Back: N/A. It can.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303029-->
+END%%
+
+%%ANKI
+Basic
+In terms of the height $h$ of a BST, what is the runtime for finding a node's successor?
+Back: $O(h)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303030-->
+END%%
+
+### Predecessors
+
+The **predecessor** of a node in a binary search tree is the node whose value would appear immediately before in an in-order traversal.
+
+%%ANKI
+Basic
+How do we define the predecessor of a BST node in terms of in-order traversals?
+Back: As the node encountered immediately before in an in-order traversal.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623611-->
+END%%
+
+%%ANKI
+Basic
+Which node is the predecessor of $7$ in the following BST?
+![[binary-search-tree.png]]
+Back: $6$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623613-->
+END%%
+
+%%ANKI
+Basic
+Which node is the predecessor of $2$ in the following BST?
+![[binary-search-tree.png]]
+Back: N/A.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623614-->
+END%%
+
+%%ANKI
+Basic
+Which node is the predecessor of $5$ in the following BST?
+![[binary-search-tree.png]]
+Back: Either $2$ or $5$.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623615-->
+END%%
+
+%%ANKI
+Basic
+Which node(s) in a BST have no predecessor?
+Back: The minimum.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623616-->
+END%%
+
+%%ANKI
+Basic
+What two cases are considered when finding the predecessor of a BST node?
+Back: If the node has a left subtree or not.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623617-->
+END%%
+
+%%ANKI
+Basic
+Suppose a BST node has a left subtree. What is its predecessor?
+Back: The maximum of the left subtree.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623618-->
+END%%
+
+%%ANKI
+Basic
+Suppose a BST node does not have a left subtree. What is its predecessor?
+Back: The first proper ancestor reached from the RHS.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722709623619-->
+END%%
+
+%%ANKI
+Cloze
+If a BST node has a left subtree, it's predecessor cannot have a {right} subtree.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303031-->
+END%%
+
+%%ANKI
+Basic
+Suppose a BST node has a left subtree. *Why* can't its successor have a left subtree?
+Back: N/A. It can.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303033-->
+END%%
+
+%%ANKI
+Basic
+Suppose a BST node has a left subtree. *Why* can't its predecessor have a right subtree?
+Back: Because then a node in that right subtree would be the actual predecessor.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303034-->
+END%%
+
+%%ANKI
+Basic
+In terms of the height $h$ of a BST, what is the runtime for finding a node's predecessor?
+Back: $O(h)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303035-->
+END%%
+
+## Deletions
+
+Consider deleting node $z$ from a BST. There are three conceptual cases to consider corresponding to the number of children $z$ has:
+
+* If $z$ has no children we can just replace $z$ with `NIL`.
+* If $z$ has one child, we can replace $z$ with its child.
+* If $z$ has two children, we swap $z$ with either its predecessor or successor, updating pointers as necessary to maintain the binary-search-tree property.
+
+%%ANKI
+Basic
+Insert node $7.5$ into the following BST. Where is the new node located?
+![[binary-search-tree.png]]
+Back: As $8$'s left child.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303036-->
+END%%
+
+%%ANKI
+Basic
+Insert node $6.5$ into the following BST. Where is the new node located?
+![[binary-search-tree.png]]
+Back: As $7$'s left child.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303037-->
+END%%
+
+%%ANKI
+Basic
+Insert node $5.5$ into the following BST. Where is the new node located?
+![[binary-search-tree.png]]
+Back: As the lower $5$'s right child.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303038-->
+END%%
+
+%%ANKI
+Basic
+When does insertion into a BST modify the root node?
+Back: When the tree being inserted into is empty.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303039-->
+END%%
+
+%%ANKI
+Basic
+In terms of the height $h$ of a BST, what is the runtime for inserting a node?
+Back: $O(h)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303040-->
+END%%
+
+%%ANKI
+Basic
+How many cases are there to consider when deleting a node from a BST?
+Back: Three.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303041-->
+END%%
+
+%%ANKI
+Basic
+What corresponds to the cases to consider when deleting a node from a BST?
+Back: The number of children the node in question has.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303042-->
+END%%
+
+%%ANKI
+Basic
+Delete BST node $z$ with no children. Which node is $z$ replaced with?
+Back: `NIL`
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303043-->
+END%%
+
+%%ANKI
+Basic
+Delete BST node $z$ with one child. Which node is $z$ replaced with?
+Back: Its one child.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303044-->
+END%%
+
+%%ANKI
+Basic
+Delete BST node $z$ with two children. Which node is $z$ replaced with?
+Back: Either its successor or predecessor.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303045-->
+END%%
+
+%%ANKI
+Basic
+Delete BST node $z$ with two children. If replacing with its successor, what two subcases need to be considered?
+Back: If $z$'s successor is its right child or not.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303046-->
+END%%
+
+%%ANKI
+Basic
+Delete BST node $z$ with two children. If replacing with its predecessor, what two subcases need to be considered?
+Back: If $z$'s predessor is its left child or not.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303047-->
+END%%
+
+%%ANKI
+Basic
+In terms of the height $h$ of a BST, what is the runtime for deleting a node?
+Back: $O(h)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303048-->
+END%%
+
+%%ANKI
+Basic
+Delete $z$ from the following BST. What does the resulting tree look like?
+![[bst-right-child.png]]
+Back:
+![[bst-right-child-after.png]]
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303049-->
+END%%
+
+%%ANKI
+Basic
+Delete $z$ from the following BST. What does the resulting tree look like?
+![[bst-left-child.png]]
+Back:
+![[bst-left-child-after.png]]
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303050-->
+END%%
+
+%%ANKI
+Basic
+Delete $z$ from the following BST. What does the resulting tree look like?
+![[bst-right-succ.png]]
+Back:
+![[bst-right-succ-after.png]]
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303051-->
+END%%
+
+%%ANKI
+Basic
+Delete $z$ from the following BST. What does the resulting tree look like?
+![[bst-deep-succ.png]]
+Back:
+![[bst-deep-succ-after.png]]
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713303052-->
+END%%
+
+%%ANKI
+Basic
+What arbitrary choice was implied in the deletion algorithm of the following BST?
+![[bst-deep-succ.png]]
+Back: To replace deleted nodes with their successor instead of predecessor.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1722713414719-->
+END%%
+
 ## Bibliography
 
 * Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

notes/hashing/index.md

@@ -328,7 +328,7 @@ END%%
 %%ANKI
 Basic
 Let $h$ be a division method hash function. What does $h(10)$ evaluate to?
-Back: To $10 \bmod{m}$, where $m$ is the number of slots in the hash table.
+Back: $10 \bmod{m}$, where $m$ is the number of slots in the hash table.
 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 Tags: hashing::static
 <!--ID: 1720889385419-->

notes/ontology/rdf/index.md

@@ -123,7 +123,7 @@ END%%
 
 %%ANKI
 Basic
-RDF identifiers are represented using what?
+RDF identifiers use what encoding?
 Back: URIs.
 Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
 <!--ID: 1722191359882-->

notes/ontology/rdf/uri.md

@@ -67,7 +67,7 @@ where the `<authority>` is further composed as
 Cloze
 A URI has the following generic syntax:
 
-{`<schema>`}`:`{`[//<authority>]`}`]`{`<path>`}{`[?<query>]`}{`[#<fragment>]`}.
+{`<scheme>`}`:`{`[//<authority>]`}{`<path>`}{`[?<query>]`}{`[#<fragment>]`}.
 Reference: “Uniform Resource Identifier.” In _Wikipedia_, July 22, 2024. [https://en.wikipedia.org/w/index.php?title=Uniform_Resource_Identifier](https://en.wikipedia.org/w/index.php?title=Uniform_Resource_Identifier&oldid=1235957234).
 <!--ID: 1722211276499-->
 END%%
@@ -488,7 +488,7 @@ A **compact URI** (CURIE) is a denser representation of URIs. In its simplest fo
 %%ANKI
 Basic
 URIs in RDF are usually represented in what condensed format?
-Back: CURIE.
+Back: CURIEs.
 Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
 <!--ID: 1722191359875-->
 END%%
@@ -496,7 +496,7 @@ END%%
 %%ANKI
 Basic
 What is CURIE an acronym for?
-Back: **C**ompact **URI**&.
+Back: **C**ompact **URI**s.
 Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
 <!--ID: 1722191359876-->
 END%%

notes/set/relations.md

@@ -647,7 +647,7 @@ END%%
 %%ANKI
 Basic
 Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive on $a$?
-Back: N/A. We must ask if $R$ is reflexive on a set.
+Back: N/A. We should ask if $R$ is reflexive on set $\{a\}$.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1720967429808-->
 END%%
@@ -729,7 +729,7 @@ END%%
 %%ANKI
 Basic
 Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ irreflexive on $a$?
-Back: N/A. We must ask if $R$ is irreflexive on a set.
+Back: N/A. We should ask if $R$ is irreflexive on set $\{a\}$.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1721870888391-->
 END%%