Add positional tree notes.
parent
f3b59d77f6
commit
45bcc6744c
|
|
@ -148,7 +148,7 @@
|
|||
"_journal/2024-02-02.md": "a3b222daee8a50bce4cbac699efc7180",
|
||||
"_journal/2024-02-01.md": "3aa232387d2dc662384976fd116888eb",
|
||||
"_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",
|
||||
"_journal/2024-04/2024-04-08.md": "764b40fc657740d024b54ddd51ffb153",
|
||||
"_journal/2024-04/2024-04-07.md": "848b6c8702e799c38e4f5dc506891bf7",
|
||||
"_journal/2024-04/2024-04-06.md": "14482b513cbe15904f1b8d81b5a143d0",
|
||||
"_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": [
|
||||
|
|
|
|||
|
|
@ -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)
|
||||
|
|
@ -8,4 +8,6 @@ title: "2024-04-13"
|
|||
- [ ] Go (1 Life & Death Problem)
|
||||
- [ ] Korean (Read 1 Story)
|
||||
- [ ] Interview Prep (1 Practice Problem)
|
||||
- [x] Log Work Hours (Max 3 hours)
|
||||
- [x] Log Work Hours (Max 3 hours)
|
||||
|
||||
* Play tested the hide-and-seek application
|
||||
|
|
@ -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
|
||||
Considered as rooted trees, are the following trees the same?
|
||||
![[ordered-binary-tree-cmp.png]]
|
||||
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: 1712409466660-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Considered as ordered trees, are the following trees the same?
|
||||
![[ordered-binary-tree-cmp.png]]
|
||||
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).
|
||||
<!--ID: 1712409466670-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Considered as binary trees, are the following trees the same?
|
||||
![[ordered-binary-tree-cmp.png]]
|
||||
Back: No.
|
||||
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).
|
||||
<!--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).
|
||||
* Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
||||
|
|
|
|||
Loading…
Reference in New Issue