Add positional tree notes.
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"_journal/2024-02-02.md": "a3b222daee8a50bce4cbac699efc7180",
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"_journal/2024-02-01.md": "3aa232387d2dc662384976fd116888eb",
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"_journal/2024-01-31.md": "7c7fbfccabc316f9e676826bf8dfe970",
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"logic/equiv-trans.md": "1198fbea260527e7dfd17b72955f5bee",
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"logic/equiv-trans.md": "4de3b5412f29b080a5f9f3629db38389",
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"_journal/2024-02-07.md": "8d81cd56a3b33883a7706d32e77b5889",
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"algorithms/loop-invariants.md": "cbefc346842c21a6cce5c5edce451eb2",
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"algorithms/loop-invariant.md": "3b390e720f3b2a98e611b49a0bb1f5a9",
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@ -310,7 +310,7 @@
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"x86-64/declarations.md": "75bc7857cf2207a40cd7f0ee056af2f2",
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"x86-64/instructions.md": "240b4ceddf174f48207ba6bed4d25246",
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"git/refs.md": "954fc69004aa65b358ec5ce07c1435ce",
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"set/trees.md": "fd092c77c26d0b99b3131da6dc043edd",
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"set/trees.md": "0e2d146f7e952aa4575e4daa82301ecd",
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"_journal/2024-03-24.md": "1974cdb9fc42c3a8bebb8ac76d4b1fd6",
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"_journal/2024-03/2024-03-23.md": "ad4e92cc2bf37f174a0758a0753bf69b",
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"_journal/2024-03/2024-03-22.md": "a509066c9cd2df692549e89f241d7bd9",
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@ -334,13 +334,16 @@
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"_journal/2024-04-06.md": "1e3e0aa30b92987e6d53d5a366963b95",
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"_journal/2024-04/2024-04-11.md": "5baeb12bc5cc5492af766def8f849fb4",
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"_journal/2024-04-14.md": "b51ad215e566bd96617d2817484232eb",
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"_journal/2024-04/2024-04-13.md": "92db6bcb162a85fb0a4afd51999b6318",
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"_journal/2024-04/2024-04-12.md": "1ec32792d88d470c2d517e5caf64fd6f"
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},
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"fields_dict": {
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"Basic": [
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@ -0,0 +1,11 @@
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---
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title: "2024-04-14"
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---
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- [ ] Anki Flashcards
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- [ ] KoL
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- [ ] Sheet Music (10 min.)
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- [ ] Go (1 Life & Death Problem)
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- [ ] Korean (Read 1 Story)
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- [ ] Interview Prep (1 Practice Problem)
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- [ ] Log Work Hours (Max 3 hours)
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@ -9,3 +9,5 @@ title: "2024-04-13"
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- [ ] Korean (Read 1 Story)
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- [ ] Interview Prep (1 Practice Problem)
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- [x] Log Work Hours (Max 3 hours)
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* Play tested the hide-and-seek application
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@ -121,6 +121,8 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
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<!--ID: 1711136844943-->
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END%%
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## Rooted Trees
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A **rooted tree** is a free tree in which one vertex is distinguished/blessed as the **root**. We call vertices of rooted trees **nodes**.
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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**.
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@ -596,6 +598,8 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
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<!--ID: 1711136845240-->
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END%%
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### Ordered Trees
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An **ordered tree** is a rooted tree in which the children of each node are ordered.
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%%ANKI
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@ -656,6 +660,35 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
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<!--ID: 1712407152763-->
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END%%
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%%ANKI
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Basic
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Considered as rooted trees, are the following trees the same?
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![[ordered-binary-tree-cmp.png]]
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Back: Yes.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1712409466660-->
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END%%
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%%ANKI
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Basic
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Considered as ordered trees, are the following trees the same?
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![[ordered-binary-tree-cmp.png]]
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Back: Yes.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1712409466670-->
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END%%
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%%ANKI
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Basic
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Considered as binary trees, are the following trees the same?
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![[ordered-binary-tree-cmp.png]]
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Back: No.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1712409466676-->
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END%%
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### Binary Trees
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A **binary tree** $T$ is a structure defined on a finite set of nodes that either
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* contains no nodes, or
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@ -780,30 +813,21 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
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END%%
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%%ANKI
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Basic
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Considered as rooted trees, are the following trees the same?
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![[ordered-binary-tree-cmp.png]]
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Is a binary tree a $k$-ary tree?
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Back: Yes.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1712409466660-->
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END%%
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%%ANKI
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Basic
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Considered as ordered trees, are the following trees the same?
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![[ordered-binary-tree-cmp.png]]
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ANKI%%
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Is a $k$-ary tree a positional tree?
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Back: Yes.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1712409466670-->
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END%%
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%%ANKI
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Basic
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Considered as binary trees, are the following trees the same?
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![[ordered-binary-tree-cmp.png]]
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Back: No.
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ANK%%
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Is a positional tree a $k$-ary tree?
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Back: Not necessarily.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1712409466676-->
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END%%
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%%ANKI
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@ -815,6 +839,231 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
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<!--ID: 1712409466682-->
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END%%
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### Positional Trees
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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.
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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 $k$.
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%%ANKI
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Basic
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What does it mean for a binary tree to be full?
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Back: Each node has $0$ or $2$ children.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713118128213-->
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END%%
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%%ANKI
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What does it mean for a binary tree to be complete?
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Back: Each leaf has the same depth and all internal nodes have degree $2$.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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END%%
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%%ANKI
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What degrees are permitted in a full binary tree?
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Back: $0$ or $2$
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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END%%
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%%ANKI
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What degrees are permitted in a complete binary tree?
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Back: $0$ or $2$
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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END%%
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%%ANKI
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What category of rooted tree does a binary tree fall under?
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Back: A positional tree or $k$-ary tree.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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END%%
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%%ANKI
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Basic
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What distinguishes a positional tree from a $k$-ary tree?
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Back: A $k$-ary tree is a positional tree in which each node has at most $k$ children.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713118128216-->
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END%%
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%%ANKI
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Basic
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What distinguishes positional trees from ordered trees?
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Back: The same children in different positions is considered distinct in the former.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713118128219-->
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END%%
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%%ANKI
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Basic
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What is a positional tree?
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Back: A rooted tree in which each child is labeled with a specific positive integer.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713118128220-->
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END%%
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%%ANKI
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Basic
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What is a $k$-ary tree?
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Back: A positional tree in which each node has at most $k$ children.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713118128223-->
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END%%
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%%ANKI
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Basic
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Which of positional trees or $k$-ary trees are more general?
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Back: The positional tree.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713118128225-->
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END%%
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%%ANKI
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Basic
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Which of positional trees or $k$-ary trees are more general?
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Back: The positional tree.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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END%%
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%%ANKI
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Basic
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Is a binary tree a positional tree?
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Back: Yes.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713118128227-->
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END%%
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%%ANKI
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Is the concept of fullness related to positional trees or $k$-ary trees?
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Back: $k$-ary trees.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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END%%
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%%ANKI
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Basic
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Is the concept of completeness related to positional trees or $k$-ary trees?
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Back: $k$-ary trees.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713118128229-->
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END%%
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%%ANKI
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Basic
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What does it mean for a $k$-ary tree to be full?
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Back: Each node has $0$ or $k$ children.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713118128231-->
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END%%
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%%ANKI
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Basic
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What degrees are permitted in a full $k$-ary tree?
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Back: $0$ or $k$
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713118128233-->
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END%%
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%%ANKI
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Basic
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What degrees are permitted in a complete $k$-ary tree?
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Back: $0$ or $k$
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713118128234-->
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END%%
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%%ANKI
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Basic
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What does it mean for a $k$-ary tree to be complete?
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Back: All leaves have the same depth and all internal nodes have degree $k$.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713118128236-->
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END%%
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%%ANKI
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Basic
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What is the degree of an internal node in a complete $k$-ary tree'?
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Back: $k$
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713118128239-->
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END%%
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%%ANKI
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Basic
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What is the degree of an external node in a complete $k$-ary tree'?
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Back: $0$
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713118128241-->
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END%%
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%%ANKI
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Basic
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How many levels exist in a rooted tree of height $h$?
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Back: $h + 1$
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713118128242-->
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END%%
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%%ANKI
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Basic
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What is the height of a rooted tree with $k$ levels?
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Back: $k - 1$
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713118128244-->
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END%%
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%%ANKI
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Basic
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What recursive definition describes the number of nodes in each levelof a complete $k$-ary tree?
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Back: $a_n = k \cdot a_{n-1}$ with $a_0 = 1$
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: algebra::sequence
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<!--ID: 1713118128246-->
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END%%
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%%ANKI
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Basic
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What recursive definition describes the number of nodes in each level of a complete $k$-ary tree?
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Back: $a_n = k \cdot a_{n-1}$ with $a_0 = 1$
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: algebra::sequence
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<!--ID: 1713118128248-->
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END%%
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%%ANKI
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Basic
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What closed formula details the number of nodes in a complete $k$-ary tree of height $h$?
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Back: $$\frac{1 - k^h}{1 - k}$$
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: algebra::sequence
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<!--ID: 1713118128249-->
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END%%
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%%ANKI
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Basic
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What kind of sequence describes the number of nodes in a complete $k$-ary tree?
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Back: A geometric sequence.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: algebra::sequence
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<!--ID: 1713118128251-->
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END%%
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%%ANKI
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Basic
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What is the common ratio in the geometric sequence counting nodes of a complete $k$-ary tree?
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Back: $k$
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: algebra::sequence
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<!--ID: 1713118128253-->
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END%%
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%%ANKI
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Basic
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How many nodes are in a complete binary tree of height $h$?
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Back: $$\frac{1 - 2^h}{1 - 2} = 2^h - 1$$
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: algebra::sequence
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<!--ID: 1713118128255-->
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END%%
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## Bibliography
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* Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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