aarjaneiro/planetmath huggingface dataset:
polymatroid

polymatroid

The

polymatroid

defined by a given

matroid

(

E

,

r

)

is the set of
all functions

w

:

E

→

ℝ

such that

w

⁢

(

e

)

≥

0

for all

⁢

e

∈

E

∑

e

∈

S

w

⁢

(

e

)

≤

r

⁢

(

S

)

for all

⁢

S

⊂

E

.

Polymatroids are related to the

convex polytopes

seen in linear programming,
and have

similar

uses.

Title

polymatroid

Canonical

name

Polymatroid

Date of creation

2013-03-22 13:56:43

Last modified on

2013-03-22 13:56:43

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

4

Author

mathcam (2727)

Entry type

Definition

Classification

msc 05B35

Generated on Thu Feb 8 20:39:47 2018 by

LaTeXML
polyomino

polyomino

A

polyomino

consists of a number of identical

connected

squares placed
in distinct locations in the plane so that at least one side of each
square is

adjacent

to (i.e. completely coincides with the side of)
another square (if the polyomino consists of at least two squares).

A polyomino with

n

squares is called an

n-omino

. For small

n

,
polyominoes have special names. A 1-omino is called a

monomino

,
a 2-omino a

domino

, a 3-omino a

tromino

or

triomino

,
etc. The famous Tetris video game derives its name from the fact that
the bricks are

tetrominoes

or 4-ominoes.

Figure 1:

All distinct 1-, 2-, 3-, 4-, and 5-ominoes. Pentominoes
have been scaled in the figure to fit on the page.

Fixed polyominoes

(which are also called

lattice animals

)
are considered distinct if they cannot be translated into each other,
while

free polyominoes

must also be distinct under

rotation

and

reflection

.

Figure 2:

All distinct, fixed dominoes and trominoes.

The topic of how many distinct (free or fixed) n-ominoes exist for a given

n

has been the subject of much research. It is known that the
number of free n-ominoes

A

n

grows exponentially. More precisely, it
can be proven that

3.72

n

<

A

n

<

4.65

n

.

Polyominoes are special instances of

polyforms

.

Title

polyomino

Canonical name

Polyomino

Date of creation

2013-03-22 15:20:18

Last modified on

2013-03-22 15:20:18

Owner

s0 (9826)

Last modified by

s0 (9826)

Numerical id

10

Author

s0 (9826)

Entry type

Definition

Classification

msc 05B50

Defines

n-omino

Defines

domino

Defines

tromino

Defines

tetromino

Defines

fixed polyomino

Defines

lattice animal

Generated on Thu Feb 8 20:39:59 2018 by

LaTeXML
AVL tree

AVL tree

An

AVL tree

is a balanced

binary search tree

where the height of the two

subtrees

(children) of a node differs by at most one. Look-up,

insertion

, and

deletion

are

O

⁢

(

ln

⁡

n

)

, where

n

is the number of nodes in the tree.

The structure is named for the inventors, Adelson-Velskii and Landis (1962).

Title

AVL tree

Canonical name

AVLTree

Date of creation

2013-03-22 13:24:39

Last modified on

2013-03-22 13:24:39

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

9

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 05C05

Generated on Thu Feb 8 20:40:01 2018 by

LaTeXML
antichain

antichain

A subset

A

of a poset

(

P

,

<

P

)

is an

antichain

if no two

elements

are

comparable

. That is, if

a

,

b

∈

A

then

a

≮

P

b

and

b

≮

P

a

.

A

maximal antichain

of

T

is one which is maximal.

In particular, if

(

P

,

<

P

)

is a tree then the maximal antichains are exactly those antichains which

intersect

every branch, and if the tree is splitting then every level is a maximal antichain.

Title

antichain

Canonical name

Antichain

Date of creation

2013-03-22 12:52:25

Last modified on

2013-03-22 12:52:25

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Definition

Classification

msc 05C05

Classification

msc 03E05

Related topic

TreeSetTheoretic

Related topic

Aronszajn

Defines

antichain

Defines

maximal antichain

Generated on Thu Feb 8 20:40:03 2018 by

LaTeXML
balanced tree

balanced tree

A

balanced tree

is a

rooted tree

where no leaf is much farther
away from the root than any other leaf.
Different balancing allow different definitions of ”much farther” and
different amounts of work to keep them balanced. For an example, see

binary tree

.

Title

balanced tree

Canonical name

BalancedTree

Date of creation

2013-03-22 12:29:11

Last modified on

2013-03-22 12:29:11

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Data Structure

Classification

msc 05C05

Related topic

Tree

Related topic

BalancedBinaryTree

Related topic

Heap

Related topic

BinaryTree

Generated on Thu Feb 8 20:40:05 2018 by

LaTeXML
binary tree

binary tree

A

binary tree

is an ordered

rooted tree

where every node has two or fewer children. A

balanced binary tree

is a binary tree that is also a

balanced tree

.
For example,

Generated on Thu Feb 8 20:40:10 2018 by

LaTeXML
branch

branch

A subset

B

of a tree

(

T

,

<

T

)

is a

branch

if

B

is a maximal

linearly ordered

subset of

T

. That is:

•

<

T

is a linear ordering of

B

•

If

t

∈

T

∖

B

then

B

∪

{

t

}

is not linearly ordered by

<

T

.

This is the same as the intuitive conception of a branch: it is a set of nodes starting at the root and going all the way to the tip (in

infinite sets

the conception is more complicated, since there may not be a tip, but the idea is the same). Since branches are maximal there is no way to add an

element

to a branch and have it remain a branch.

A

cofinal branch

is a branch which

intersects

every level of the tree.

Title

branch

Canonical name

Branch

Date of creation

2013-03-22 12:52:22

Last modified on

2013-03-22 12:52:22

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Definition

Classification

msc 05C05

Classification

msc 03E05

Related topic

TreeSetTheoretic

Related topic

ExampleOfTreeSetTheoretic

Defines

branch

Defines

cofinal branch

Generated on Thu Feb 8 20:40:12 2018 by

LaTeXML
child node (of a tree)

child node (of a tree)

\xyoption

all

A

child node

C

of a node

P

in a tree is any node

connected

to

P

which has a path

distance

from the root node

R

which is one greater than the path distance between

P

and

R

.

Drawn in the

canonical

root-at-top manner, a child node of a node

P

in a tree is simply any node immediately below

P

which is connected to it.

\xymatrix

⁢

&

∙

\ar

⁢

@

-

[

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

&

⁢

&

∙

&

⁢

&

∙

\ar

⁢

@

-

[

d

⁢

r

]

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

&

⁢

&

∙

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

&

∙

&

∙

&

⁢

&

⁢

&

⁢

&

Figure: A node (blue) and its children (red.)

Title

child node (of a tree)

Canonical name

ChildNodeofATree

Date of creation

2013-03-22 12:30:37

Last modified on

2013-03-22 12:30:37

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

4

Author

akrowne (2)

Entry type

Definition

Classification

msc 05C05

Synonym

child node

Synonym

child

Related topic

ParentNodeInATree

Generated on Thu Feb 8 20:40:14 2018 by

LaTeXML
complete binary tree

complete binary tree

A

complete binary tree

is a

binary tree

with the additional

property

that every node must have exactly two “

children

” if an

internal node

, and zero children if a

leaf node

.

More precisely: for our

base case

, the complete binary tree of exactly one node is simply the tree consisting of that node by itself. The property of being “

complete

” is preserved if, at each step, we expand the tree by connecting exactly zero or two

individual

nodes (or complete binary trees) to any node in the tree (but both must be

connected

to the same node.)

Title

complete binary tree

Canonical

name

CompleteBinaryTree

Date of creation

2013-03-22 12:30:14

Last modified on

2013-03-22 12:30:14

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

7

Author

akrowne (2)

Entry type

Definition

Classification

msc 05C05

Synonym

complete

Related topic

ExtendedBinaryTree

Generated on Thu Feb 8 20:40:15 2018 by

LaTeXML
digital search tree

digital search tree

A

digital search tree

is a tree which stores strings internally so that there is no need for extra

leaf nodes

to store the strings.

Title

digital search tree

Canonical name

DigitalSearchTree

Date of creation

2013-03-22 12:28:31

Last modified on

2013-03-22 12:28:31

Owner

Logan (6)

Last modified by

Logan (6)

Numerical id

8

Author

Logan (6)

Entry type

Data Structure

Classification

msc 05C05

Classification

msc 68P10

Classification

msc 68P20

Related topic

DigitalTree

Generated on Thu Feb 8 20:40:17 2018 by

LaTeXML
example of tree (set theoretic)

example of tree (set theoretic)

The set

ℤ

+

is a tree with

<

T

⁣

=

⁣

<

. This isn’t a very interesting tree, since it simply consists of a line of nodes. However note that the height is

ω

even though no particular node has that height.

A more interesting tree using

ℤ

+

defines

m

<

T

n

if

i

a

=

m

and

i

b

=

n

for some

i

,

a

,

b

∈

ℤ

+

∪

{

0

}

. Then

1

is the root, and all numbers which are not powers of another number are in

T

1

. Then all squares (which are not also fourth powers) for

T

2

, and so on.

To illustrate the

concept

of a

cofinal branch

, observe that for any

limit ordinal

κ

we can construct a

κ

-tree which has no cofinal branches. We let

T

=

{

(

α

,

β

)

|

α

<

β

<

κ

}

and

(

α

1

,

β

1

)

<

T

(

α

2

,

β

2

)

↔

α

1

<

α

2

∧

β

1

=

β

2

. The tree then has

κ

disjoint

branches, each consisting of the set

{

(

α

,

β

)

|

α

<

β

}

for some

β

<

κ

. No branch is

cofinal

, since each branch is capped at

β

elements

, but for any

γ

<

κ

, there is a branch of height

γ

+

1

. Hence the

supremum

of the heights is

κ

.

Title

example of tree (set theoretic)

Canonical name

ExampleOfTreesetTheoretic

Date of creation

2013-03-22 12:52:27

Last modified on

2013-03-22 12:52:27

Owner

uzeromay (4983)

Last modified by

uzeromay (4983)

Numerical id

5

Author

uzeromay (4983)

Entry type

Example

Classification

msc 05C05

Classification

msc 03E05

Related topic

CofinalBranch

Generated on Thu Feb 8 20:40:19 2018 by

LaTeXML
extended binary tree

extended binary tree

An

extended binary tree

is a

transformation

of any

binary tree

into a

complete binary tree

. This transformation consists of replacing every

null subtree

of the original tree with “special nodes.” The nodes from the original tree are then

internal nodes

, while the “special nodes” are

external nodes

.

For

instance

, consider the following binary tree.

The following tree is its extended binary tree. Empty circles represent internal nodes, and filled circles represent external nodes.

Every internal node in the extended tree has exactly two

children

, and every external node is a leaf. The result is a complete binary tree.

Title

extended binary tree

Canonical name

ExtendedBinaryTree

Date of creation

2013-03-22 12:31:36

Last modified on

2013-03-22 12:31:36

Owner

aoh45 (5079)

Last modified by

aoh45 (5079)

Numerical id

8

Author

aoh45 (5079)

Entry type

Data Structure

Classification

msc 05C05

Related topic

BinaryTree

Related topic

CompleteBinaryTree

Related topic

ExternalPathLength

Related topic

WeightedPathLength

Related topic

MinimumWeightedPathLength

Defines

external node

Defines

internal node

Generated on Thu Feb 8 20:40:21 2018 by

LaTeXML
external path length

external path length

Given a

binary tree

T

, construct its

extended binary tree

T

′

.
The

external path length

of

T

is then defined to be the sum of the lengths of the paths to each of the external nodes.

For example, let

T

be the following tree.

The extended binary tree of

T

is

The external path length of

T

(denoted

E

) is

E

=

2

+

3

+

3

+

3

+

3

+

3

+

3

=

20

The

internal path length

of

T

is defined to be the sum of the lengths of the paths to each of the

internal nodes

. The internal path length of our example tree (denoted

I

) is

I

=

1

+

2

+

0

+

2

+

1

+

2

=

8

Note that in this case

E

=

I

+

2

⁢

n

, where

n

is the number of internal nodes. This happens to hold for all binary trees.

Title

external path length

Canonical

name

ExternalPathLength

Date of creation

2013-03-22 12:32:03

Last modified on

2013-03-22 12:32:03

Owner

Logan (6)

Last modified by

Logan (6)

Numerical id

4

Author

Logan (6)

Entry type

Definition

Classification

msc 05C05

Related topic

ExtendedBinaryTree

Related topic

WeightedPathLength

Defines

external path length

Defines

internal path length

Generated on Thu Feb 8 20:40:23 2018 by

LaTeXML
internal node (of a tree)

internal node (of a tree)

\xyoption

all

An

internal node

of a tree is any node which has degree greater than one. Or, phrased in

rooted tree

terminology, the internal nodes of a tree are the nodes which have at least one

child node

.

\xymatrix

⁢

&

∙

\ar

⁢

@

-

[

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

&

⁢

&

∙

&

⁢

&

∙

\ar

⁢

@

-

[

d

⁢

r

]

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

&

⁢

&

∙

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

&

∙

&

∙

&

⁢

&

⁢

&

⁢

&

Figure: A tree with internal nodes highlighted in red.

Title

internal node (of a tree)

Canonical

name

InternalNodeofATree

Date of creation

2013-03-22 12:30:25

Last modified on

2013-03-22 12:30:25

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

6

Author

akrowne (2)

Entry type

Definition

Classification

msc 05C05

Synonym

internal node

Generated on Thu Feb 8 20:40:25 2018 by

LaTeXML
König’s lemma

König’s lemma

Theorem

(König’s lemma).

Let

T

be a rooted directed tree. If each vertex has finite
degree but there are arbitrarily long rooted paths in

T

,
then

T

contains an

infinite

path.

Proof.

For each

n

≥

1

, let

P

n

be a rooted path in

T

of length

n

,
and let

c

n

be the child of the root appearing in

P

n

.
By

assumption

, the set

{

c

n

∣

n

≥

1

}

is finite.
Since the set

{

P

n

∣

n

≥

1

}

is infinite, the

pigeonhole principle

implies that there is a child

c

of the root
such that

c

=

c

n

for infinitely many

n

.

Now let us look at the

subtree

T

c

of

T

rooted at

c

. Each vertex
has finite degree, and since there are paths

P

n

of arbitrarily
long length in

T

passing through

c

, there are arbitrarily long
paths in

T

c

rooted at

c

. Hence if

T

satisfies

the

hypothesis

of the lemma, the root has a child

c

such that

T

c

also satisfies
the hypothesis of the lemma. Hence we may inductively build up a
path in

T

of infinite length, at each stage selecting a child so
that the subtree rooted at that vertex still has arbitrarily long
paths.
∎

References

1

Kleene, Stephen.,

Mathematical Logic

, New York: Wiley, 1967.

Title

König’s lemma

Canonical name

KonigsLemma

Date of creation

2013-03-22 15:38:26

Last modified on

2013-03-22 15:38:26

Owner

mps (409)

Last modified by

mps (409)

Numerical id

6

Author

mps (409)

Entry type

Theorem

Classification

msc 05C05

Synonym

Koenig’s lemma

Generated on Thu Feb 8 20:40:27 2018 by

LaTeXML
leaf node (of a tree)

leaf node (of a tree)

\xyoption

all

A

leaf

of a tree is any node which has degree of exactly 1. Put another way, a

leaf node

of a

rooted tree

is any node which has no

child nodes

.

\xymatrix

⁢

&

∙

\ar

⁢

@

-

[

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

&

⁢

&

∙

&

⁢

&

∙

\ar

⁢

@

-

[

d

⁢

r

]

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

&

⁢

&

∙

\ar

⁢

@

-

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d

⁢

l

]

⁢

&

⁢

&

∙

&

∙

&

⁢

&

⁢

&

⁢

&

Figure: A tree with leaf nodes highlighted in red.

Title

leaf node (of a tree)

Canonical

name

LeafNodeofATree

Date of creation

2013-03-22 12:30:28

Last modified on

2013-03-22 12:30:28

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

5

Author

akrowne (2)

Entry type

Definition

Classification

msc 05C05

Synonym

leaf node

Synonym

leaf

Generated on Thu Feb 8 20:40:29 2018 by

LaTeXML
minimum spanning tree

minimum spanning tree

Given a graph

G

with weighted edges, a

minimum spanning tree

is a

spanning tree

with minimum weight, where the weight of a spanning tree is the sum of the weights of its edges. There may be more than one minimum spanning tree for a graph, since it is the weight of the spanning tree that must be minimum.

For example, here is a graph

G

of weighted edges and a minimum spanning tree

T

for that graph. The edges of

T

are drawn as solid lines, while edges in

G

but not in

T

are drawn as dotted lines.

Generated on Thu Feb 8 20:40:34 2018 by

LaTeXML
minimum weighted path length

minimum weighted path length

Given a list of weights,

W

:=

{

w

1

,

w

2

,

…

,

w

n

}

, the

minimum weighted path length

is the minimum of the

weighted path length

of all

extended binary trees

that have

n

external nodes with weights taken from

W

. There may be multiple possible trees that give this minimum

path length

, and quite often finding this tree is more important than determining the path length.

Example

Let

W

:=

{

1

,

2

,

3

,

3

,

4

}

. The minimum weighted path length is

29

. A tree that gives this weighted path length is shown below.

Applications

Constructing a tree of minimum weighted path length for a given set of weights has several applications, particularly dealing with optimization problems.
A simple and elegant

algorithm

for constructing such a tree is Huffman’s algorithm.
Such a tree can give the most optimal algorithm for merging

n

sorted sequences (optimal merge). It can also provide a means of compressing data (

Huffman coding

), as well as lead to optimal searches.

Title

minimum weighted path length

Canonical name

MinimumWeightedPathLength

Date of creation

2013-03-22 12:32:12

Last modified on

2013-03-22 12:32:12

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

6

Author

mathwizard (128)

Entry type

Definition

Classification

msc 05C05

Related topic

WeightedPathLength

Related topic

ExtendedBinaryTree

Related topic

HuffmansAlgorithm

Related topic

HuffmanCoding

Generated on Thu Feb 8 20:40:36 2018 by

LaTeXML
null tree

null tree

A

null tree

is simply a tree with zero nodes.

Title

null tree

Canonical name

NullTree

Date of creation

2013-03-22 12:31:16

Last modified on

2013-03-22 12:31:16

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Data Structure

Classification

msc 05C05

Synonym

null subtree

Synonym

empty tree

Synonym

empty subtree

Related topic

Tree

Generated on Thu Feb 8 20:40:38 2018 by

LaTeXML
parent node (in a tree)

parent node (in a tree)

\xyoption

all

A

parent

node

P

of a node

C

in a tree is the first node which lies along the path from

C

to the root of the tree,

R

.

Drawn in the

canonical

root-at-top manner, the

parent node

of a node

C

in a tree is simply the node immediately above

C

which is

connected

to it.

\xymatrix

⁢

&

∙

\ar

⁢

@

-

[

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

&

⁢

&

∙

&

⁢

&

∙

\ar

⁢

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[

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⁢

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]

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

&

⁢

&

∙

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

&

∙

&

∙

&

⁢

&

⁢

&

⁢

&

Figure: A node (blue) and its parent (red.)

Title

parent node (in a tree)

Canonical name

ParentNodeinATree

Date of creation

2013-03-22 12:30:32

Last modified on

2013-03-22 12:30:32

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

5

Author

akrowne (2)

Entry type

Definition

Classification

msc 05C05

Synonym

parent node

Synonym

parent

Related topic

ChildNodeOfATree

Generated on Thu Feb 8 20:40:40 2018 by

LaTeXML
proof that ω has the tree property

proof that

ω

has the tree property

Let

T

be a tree with finite levels and an

infinite

number of

elements

. Then consider the elements of

T

0

.

T

can be partitioned into the set of

descendants

of each of these elements, and since any finite

partition

of an

infinite set

has at least one infinite partition, some element

x

0

in

T

0

has an infinite number of descendants. The same procedure can be applied to the children of

x

0

to give an element

x

1

∈

T

1

which has an infinite number of descendants, and then to the children of

x

1

, and so on. This gives a

sequence

X

=

⟨

x

0

,

x

1

,

…

⟩

. The sequence is infinite since each element has an infinite number of descendants, and since

x

i

+

1

is always of child of

x

i

,

X

is a branch, and therefore an infinite branch of

T

.

Title

proof that

ω

has the

tree property

Canonical name

ProofThatomegaHasTheTreeProperty

Date of creation

2013-03-22 12:52:36

Last modified on

2013-03-22 12:52:36

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Proof

Classification

msc 05C05

Classification

msc 03E05

Synonym

proof that omega has the tree property

Synonym

proof that

infinity

has the tree property

Generated on Thu Feb 8 20:40:41 2018 by

LaTeXML
root (of a tree)

root (of a tree)

\xyoption

all

The

root

of a tree is a place-holder node. It is typically drawn at the top of the page, with the other nodes below (with all nodes having the same path

distance

from the root at the same height.)

\xymatrix

⁢

&

∙

\ar

⁢

@

-

[

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

&

⁢

&

∙

&

⁢

&

∙

\ar

⁢

@

-

[

d

⁢

r

]

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

&

⁢

&

∙

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

&

∙

&

∙

&

⁢

&

⁢

&

⁢

&

Figure: A tree with root highlighted in red.

Any tree can be redrawn this way, selecting any node as the root. This is important to note: taken as a graph in general, the notion of “root” is meaningless. We introduce a root explicitly when we begin speaking of a graph as a tree– there is nothing in general that selects a root for us.

However, there are some special cases of trees where the root can be distinguished from the other nodes implicitly due to the

properties

of the tree. For

instance

, a root is uniquely identifiable in a

complete binary tree

, where it is the only node with degree two.

Further, in a

directed tree

, there can be only one root. This is because all nodes must be

reachable

from that node by following the arrows between them in the proper direction.

Title

root (of a tree)

Canonical

name

RootofATree

Date of creation

2013-03-22 12:30:11

Last modified on

2013-03-22 12:30:11

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

8

Author

akrowne (2)

Entry type

Definition

Classification

msc 05C05

Synonym

root

Generated on Thu Feb 8 20:40:53 2018 by

LaTeXML
spanning tree

spanning tree

A

spanning tree

of a (

connected

) graph

G

is a connected,

acyclic

subgraph

of

G

that contains all of the vertices of

G

. Below is an example of a spanning tree

T

, where the edges in

T

are drawn as solid lines and the edges in

G

but not in

T

are drawn as dotted lines.

Generated on Thu Feb 8 20:40:58 2018 by

LaTeXML
Suslin tree

Suslin tree

An

Aronszajn tree

is a

Suslin tree

iff it has no

uncountable

antichains

.

Title

Suslin tree

Canonical name

SuslinTree

Date of creation

2013-03-22 12:52:42

Last modified on

2013-03-22 12:52:42

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Definition

Classification

msc 05C05

Classification

msc 03E05

Related topic

TreeSetTheoretic

Related topic

AronszajnTree

Related topic

Aronszajn

Defines

Suslin tree

Generated on Thu Feb 8 20:41:00 2018 by

LaTeXML
tree

tree

Formally, a

forest

is an undirected,

acyclic graph

. A forest consists of

trees

, which are themselves acyclic,

connected graphs

. For example,
the following diagram represents a forest, each

connected component

of which is a tree.

Generated on Thu Feb 8 20:41:05 2018 by

LaTeXML