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Depth [expr]

gives the maximum number of indices needed to specify any part of expr, plus 1.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Depths of General Expressions  
Depths of Special Expressions  
Options  
Heads  
Applications  
Properties & Relations  
Possible Issues  
See Also
Tech Notes
Related Guides
Related Links
History
Cite this Page

Depth [expr]

gives the maximum number of indices needed to specify any part of expr, plus 1.

Details and Options

  • Raw objects have depth 1.
  • With the option setting Heads->True , Depth includes heads of expressions and their parts.

Examples

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Basic Examples  (3)

Symbols have depth 1:

Additional levels of nesting increase the depth:

Only the deepest part of the expression affects the depth:

Depth can work with any expression:

Use TreeForm to visualize depth as the number of levels of the expression:

Scope  (10)

Depths of General Expressions  (5)

Depth works on any expression, not just lists:

Depth does not distinguish heads:

Symbols and strings have depth 1:

All numbers have depth 1:

Numeric expressions that are not NumberQ have greater depth:

Depth does not by default count depth in the head:

Use the option setting Heads True to include heads:

Depths of Special Expressions  (5)

Keys are ignored when computing the depth of an association:

Compare with:

The depth of values is considered when computing the depth of an association:

Depth of a nested association:

Depth treats SparseArray and structured array objects just like the corresponding ordinary lists:

Most raw objects have a depth of 1:

Options  (1)

Heads  (1)

By default, the depth of heads is ignored:

Include the depth of heads:

Applications  (2)

Find how deeply nested results from integrals are:

Find the depths of combinator expressions [more info]:

Considering the heads increases the depth:

Increase the nesting level of the combinator expressions:

Properties & Relations  (5)

Depth generally gives the length of the maximum index, plus 1:

Verify the equality:

Depth considers the deepest part of an expression:

ArrayDepth only considers the levels to which the expression is completely rectangular:

For completely rectangular expressions, Depth gives a result one greater than ArrayDepth :

Depth [expr] is the smallest positive level k for which Level [expr,{k}] returns an empty list:

If length-0 functions or compound heads are present, both Depth and Level must use Heads True :

Successive elements from NestList have larger depths:

Possible Issues  (3)

Depth returns a depth one greater than that returned by ArrayDepth :

Depth counts an association as a single level:

It counts the corresponding list of rules as two levels:

The relationship between Depth and Position can break down when compound heads are present:

Set Heads True in Depth to restore the relationship:

See Also

ArrayDepth   Level   LeafCount   Length   Nest

History

Introduced in 1988 (1.0) | Updated in 2003 (5.0) 2010 (8.0) 2018 (11.3)

Wolfram Research (1988), Depth, Wolfram Language function, https://reference.wolfram.com/language/ref/Depth.html (updated 2018).

Text

Wolfram Research (1988), Depth, Wolfram Language function, https://reference.wolfram.com/language/ref/Depth.html (updated 2018).

CMS

Wolfram Language. 1988. "Depth." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2018. https://reference.wolfram.com/language/ref/Depth.html.

APA

Wolfram Language. (1988). Depth. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Depth.html

BibTeX

@misc{reference.wolfram_2025_depth, author="Wolfram Research", title="{Depth}", year="2018", howpublished="\url{https://reference.wolfram.com/language/ref/Depth.html}", note=[Accessed: 16-November-2025]}

BibLaTeX

@online{reference.wolfram_2025_depth, organization={Wolfram Research}, title={Depth}, year={2018}, url={https://reference.wolfram.com/language/ref/Depth.html}, note=[Accessed: 16-November-2025]}

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