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CompleteKaryTree [n]

gives the complete binary tree with n levels.

CompleteKaryTree [n,k]

gives the complete k-ary tree with n levels.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Options  
AnnotationRules  
DirectedEdges  
EdgeLabels  
Show More Show More
EdgeShapeFunction  
EdgeStyle  
EdgeWeight  
GraphLayout  
GraphHighlight  
GraphHighlightStyle  
PlotTheme  
Base Themes  
Feature Themes  
VertexCoordinates  
VertexLabels  
VertexShape  
VertexShapeFunction  
VertexSize  
VertexStyle  
VertexWeight  
Applications  
Properties & Relations  
See Also
Related Guides
History
Cite this Page

CompleteKaryTree [n]

gives the complete binary tree with n levels.

CompleteKaryTree [n,k]

gives the complete k-ary tree with n levels.

Details and Options

Examples

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Basic Examples  (3)

A complete binary tree with 5 levels:

A complete ternary tree with 3 levels:

Use directed edges:

Options  (79)

AnnotationRules  (2)

Specify an annotation for vertices:

Edges:

DirectedEdges  (1)

By default, an undirected graph is generated:

Use DirectedEdges->True to generate a directed graph:

EdgeLabels  (6)

Label the edge 12:

Label all edges individually:

Use any expression as a label:

Use explicit coordinates to place labels:

Vary positions within the label:

Place multiple labels:

Use automatic labeling by values through Tooltip and StatusArea :

EdgeShapeFunction  (6)

Get a list of built-in settings for EdgeShapeFunction :

Undirected edges including the basic line:

Lines with different glyphs on the edges:

Directed edges including solid arrows:

Line arrows:

Open arrows:

Specify an edge function for an individual edge:

Combine with a different default edge function:

Draw edges by running a program:

EdgeShapeFunction can be combined with EdgeStyle :

EdgeShapeFunction has higher priority than EdgeStyle :

EdgeStyle  (2)

Style edges:

Style individual edges:

EdgeWeight  (2)

Specify the weight for all edges:

Use any numeric expression as a weight:

GraphLayout  (5)

By default, the layout is chosen automatically:

Specify layouts on special curves:

Specify layouts that satisfy optimality criteria:

VertexCoordinates overrides GraphLayout coordinates:

Use AbsoluteOptions to extract VertexCoordinates computed using a layout algorithm:

GraphHighlight  (3)

Highlight the vertex 1:

Highlight the edge 13:

Highlight the vertices and edges:

GraphHighlightStyle  (2)

Get a list of built-in settings for GraphHighlightStyle :

Use built-in settings for GraphHighlightStyle :

PlotTheme  (4)

Base Themes  (2)

Use a common base theme:

Use a monochrome theme:

Feature Themes  (2)

Use a large graph theme:

Use a classic diagram theme:

VertexCoordinates  (3)

By default, any vertex coordinates are computed automatically:

Extract the resulting vertex coordinates using AbsoluteOptions :

Specify a layout function along an ellipse:

Use it to generate vertex coordinates for a graph:

VertexCoordinates has higher priority than GraphLayout :

VertexLabels  (13)

Use vertex names as labels:

Label individual vertices:

Label all vertices:

Use any expression as a label:

Use Placed with symbolic locations to control label placement, including outside positions:

Symbolic outside corner positions:

Symbolic inside positions:

Symbolic inside corner positions:

Use explicit coordinates to place the center of labels:

Place all labels at the upper-right corner of the vertex and vary the coordinates within the label:

Place multiple labels:

Any number of labels can be used:

Use the argument to Placed to control formatting including Tooltip :

Or StatusArea :

Use more elaborate formatting functions:

VertexShape  (5)

Use any Graphics , Image , or Graphics3D as a vertex shape:

Specify vertex shapes for individual vertices:

VertexShape can be combined with VertexSize :

VertexShape is not affected by VertexStyle :

VertexShapeFunction has higher priority than VertexShape :

VertexShapeFunction  (10)

Get a list of built-in collections for VertexShapeFunction :

Use built-in settings for VertexShapeFunction in the "Basic" collection:

Simple basic shapes:

Common basic shapes:

Use built-in settings for VertexShapeFunction in the "Rounded" collection:

Use built-in settings for VertexShapeFunction in the "Concave" collection:

Draw individual vertices:

Combine with a default vertex function:

Draw vertices using a predefined graphic:

Draw vertices by running a program:

VertexShapeFunction can be combined with VertexStyle :

VertexShapeFunction has higher priority than VertexStyle :

VertexShapeFunction can be combined with VertexSize :

VertexShapeFunction has higher priority than VertexShape :

VertexSize  (8)

By default, the size of vertices is computed automatically:

Specify the size of all vertices using symbolic vertex size:

Use a fraction of the minimum distance between vertex coordinates:

Use a fraction of the overall diagonal for all vertex coordinates:

Specify size in both the and directions:

Specify the size for individual vertices:

VertexSize can be combined with VertexShapeFunction :

VertexSize can be combined with VertexShape :

VertexStyle  (5)

Style all vertices:

Style individual vertices:

VertexShapeFunction can be combined with VertexStyle :

VertexShapeFunction has higher priority than VertexStyle :

VertexStyle can be combined with BaseStyle :

VertexStyle has higher priority than BaseStyle :

VertexShape is not affected by VertexStyle :

VertexWeight  (2)

Set the weight for all vertices:

Use any numeric expression as a weight:

Applications  (7)

The GraphCenter of a complete k-ary tree:

The GraphPeriphery :

The VertexEccentricity :

Highlight the vertex eccentricity path:

The GraphRadius :

Highlight the radius path:

The GraphDiameter :

Highlight the diameter path:

Highlight the vertex degree for CompleteKaryTree :

Highlight the closeness centrality:

Highlight the eigenvector centrality:

Vertex connectivity from to is the number of vertex independent paths from to :

The vertex connectivity for a tree is 1 for all vertex pairs:

Properties & Relations  (8)

CompleteKaryTree [n,k] has vertices:

CompleteKaryTree [n,k] has edges:

A complete k-ary tree with vertices has edges:

A complete k-ary tree is a tree graph:

A complete k-ary tree is a bipartite graph:

A complete k-ary tree is acyclic:

A complete k-ary tree is loop free:

A complete k-ary tree is a special case of a k-ary tree:

See Also

KaryTree   TreeGraph   StarGraph   Groupings

Function Repository: CompleteLevelsKaryTree

Wolfram Research (2010), CompleteKaryTree, Wolfram Language function, https://reference.wolfram.com/language/ref/CompleteKaryTree.html.

Text

Wolfram Research (2010), CompleteKaryTree, Wolfram Language function, https://reference.wolfram.com/language/ref/CompleteKaryTree.html.

CMS

Wolfram Language. 2010. "CompleteKaryTree." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CompleteKaryTree.html.

APA

Wolfram Language. (2010). CompleteKaryTree. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CompleteKaryTree.html

BibTeX

@misc{reference.wolfram_2025_completekarytree, author="Wolfram Research", title="{CompleteKaryTree}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/CompleteKaryTree.html}", note=[Accessed: 17-November-2025]}

BibLaTeX

@online{reference.wolfram_2025_completekarytree, organization={Wolfram Research}, title={CompleteKaryTree}, year={2010}, url={https://reference.wolfram.com/language/ref/CompleteKaryTree.html}, note=[Accessed: 17-November-2025]}

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