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ArrayDepth

✖
ArrayDepth[expr]

gives the depth to which expr is a full array, with all the parts at a particular level having the same length.

Details and Options

Examples

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Basic Examples  (1)Summary of the most common use cases

Find the depth to which an array is full:

In[1]:=1
✖
https://wolfram.com/xid/0cg41kau6-vln
Out[1]=1

Scope  (4)Survey of the scope of standard use cases

ArrayDepth counts only dimensions at which an expression is not "ragged":

In[108]:=108
✖
https://wolfram.com/xid/0cg41kau6-nxp
In[8]:=8
✖
https://wolfram.com/xid/0cg41kau6-co6
Out[8]=8

ArrayDepth works with special array types, including SparseArray objects:

In[1]:=1
✖
https://wolfram.com/xid/0cg41kau6-s21
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg41kau6-gy1
Out[2]=2

SymmetrizedArray objects:

In[3]:=3
✖
https://wolfram.com/xid/0cg41kau6-k8hdtp
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cg41kau6-dd5rgd
Out[4]=4

NumericArray objects:

In[5]:=5
✖
https://wolfram.com/xid/0cg41kau6-o1re6
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0cg41kau6-uwc35r
Out[6]=6

Find the depth of a matrix that is a mixture of List and different special array heads:

In[1]:=1
✖
https://wolfram.com/xid/0cg41kau6-k3u74c
Out[1]=1

ArrayDepth works with any head, not just List:

In[1]:=1
✖
https://wolfram.com/xid/0cg41kau6-vxk
Out[1]=1

Options  (4)Common values & functionality for each option

AllowedHeads  (4)

Only consider nested expressions with head List:

In[1]:=1
✖
https://wolfram.com/xid/0cg41kau6-k9dyn2
Out[1]=1

By default, rectangular nested expressions with any head are considered arrays:

In[2]:=2
✖
https://wolfram.com/xid/0cg41kau6-khkuch
Out[2]=2

Allow lists and numeric arrays at different levels:

In[1]:=1
✖
https://wolfram.com/xid/0cg41kau6-7clrof
Out[1]=1

Different heads cannot be mixed at the same level:

In[2]:=2
✖
https://wolfram.com/xid/0cg41kau6-3nvfbn
Out[2]=2

Allow a mixture of both heads at the same level:

In[3]:=3
✖
https://wolfram.com/xid/0cg41kau6-548h4q
Out[3]=3

Allow List as well as other general array representations:

In[1]:=1
✖
https://wolfram.com/xid/0cg41kau6-9ml35q
Out[1]=1

Only allow the head List:

In[2]:=2
✖
https://wolfram.com/xid/0cg41kau6-70jjg6
Out[2]=2

Both settings disallow general heads:

In[3]:=3
✖
https://wolfram.com/xid/0cg41kau6-vcu7w6
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cg41kau6-scvobz
Out[4]=4

Allow any head at each level of the array:

In[1]:=1
✖
https://wolfram.com/xid/0cg41kau6-fwqt97
Out[1]=1

An array has a consistent head at each level:

In[2]:=2
✖
https://wolfram.com/xid/0cg41kau6-xyv2j1
Out[2]=2

Completely ignore heads, looking only at the tree structure of the expression:

In[3]:=3
✖
https://wolfram.com/xid/0cg41kau6-lleebw
Out[3]=3

Properties & Relations  (8)Properties of the function, and connections to other functions

Length[Dimensions[expr]] equals ArrayDepth[expr]:

In[1]:=1
✖
https://wolfram.com/xid/0cg41kau6-t2q1lr
Out[1]=1

ArrayDepth always returns a non-negative integer:

In[1]:=1
✖
https://wolfram.com/xid/0cg41kau6-plvazy
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg41kau6-szy46a
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cg41kau6-5o5zhi
Out[3]=3

The default value of the option AllowedHeads allows mixtures of list, sparse and structured arrays:

In[1]:=1
✖
https://wolfram.com/xid/0cg41kau6-7j5w0a
Out[1]=1

It also allows any single head at successive levels:

In[2]:=2
✖
https://wolfram.com/xid/0cg41kau6-hrep0
Out[2]=2

The following setting of AllowedHeads imitates the default behavior with respect to list-like arrays:

In[3]:=3
✖
https://wolfram.com/xid/0cg41kau6-bqcoq5
Out[3]=3

However, it excludes arbitrary heads:

In[4]:=4
✖
https://wolfram.com/xid/0cg41kau6-wtahnf
Out[4]=4

VectorQ, MatrixQ and ArrayQ effectively use AllowedHeads"ListLike":

In[1]:=1
✖
https://wolfram.com/xid/0cg41kau6-pmpfyl
In[2]:=2
✖
https://wolfram.com/xid/0cg41kau6-4b78jt
In[3]:=3
✖
https://wolfram.com/xid/0cg41kau6-iksb32
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cg41kau6-165ab7
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0cg41kau6-sm8t33
Out[5]=5

The setting AllowedHeads"ListLike" does not include ByteArray and NumericArray:

In[1]:=1
✖
https://wolfram.com/xid/0cg41kau6-b4elld
In[2]:=2
✖
https://wolfram.com/xid/0cg41kau6-ieinh6
Out[2]=2

Combine "ListLike" with the additional heads to allow the combination:

In[3]:=3
✖
https://wolfram.com/xid/0cg41kau6-no0ujw
Out[3]=3

ArrayDepth[Array[h,dims]] gives Length[dims]:

In[1]:=1
✖
https://wolfram.com/xid/0cg41kau6-q2t
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0cg41kau6-1owf7w
Out[1]=1

Depth considers the deepest part of an expression:

In[2]:=2
✖
https://wolfram.com/xid/0cg41kau6-kcrjnc
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0cg41kau6-4lpecw
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg41kau6-cyhoza
Out[2]=2

Possible Issues  (1)Common pitfalls and unexpected behavior

Numeric expressions report array depth based of the structure of their FullForm:

In[1]:=1
✖
https://wolfram.com/xid/0cg41kau6-0s5y0i
In[2]:=2
✖
https://wolfram.com/xid/0cg41kau6-ymwq71
Out[2]=2

To treat all numeric expressions as scalars, use the setting List or "ListLike" for AllowedHeads:

In[3]:=3
✖
https://wolfram.com/xid/0cg41kau6-cjt00d
Out[3]=3
Wolfram Research (2003), ArrayDepth, Wolfram Language function, https://reference.wolfram.com/language/ref/ArrayDepth.html (updated 2020).
✖
Wolfram Research (2003), ArrayDepth, Wolfram Language function, https://reference.wolfram.com/language/ref/ArrayDepth.html (updated 2020).

Text

Wolfram Research (2003), ArrayDepth, Wolfram Language function, https://reference.wolfram.com/language/ref/ArrayDepth.html (updated 2020).

✖
Wolfram Research (2003), ArrayDepth, Wolfram Language function, https://reference.wolfram.com/language/ref/ArrayDepth.html (updated 2020).

CMS

Wolfram Language. 2003. "ArrayDepth." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/ArrayDepth.html.

✖
Wolfram Language. 2003. "ArrayDepth." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/ArrayDepth.html.

APA

Wolfram Language. (2003). ArrayDepth. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArrayDepth.html

✖
Wolfram Language. (2003). ArrayDepth. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArrayDepth.html

BibTeX

@misc{reference.wolfram_2025_arraydepth, author="Wolfram Research", title="{ArrayDepth}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/ArrayDepth.html}", note=[Accessed: 01-June-2025 ]}

✖
@misc{reference.wolfram_2025_arraydepth, author="Wolfram Research", title="{ArrayDepth}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/ArrayDepth.html}", note=[Accessed: 01-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_arraydepth, organization={Wolfram Research}, title={ArrayDepth}, year={2020}, url={https://reference.wolfram.com/language/ref/ArrayDepth.html}, note=[Accessed: 01-June-2025 ]}

✖
@online{reference.wolfram_2025_arraydepth, organization={Wolfram Research}, title={ArrayDepth}, year={2020}, url={https://reference.wolfram.com/language/ref/ArrayDepth.html}, note=[Accessed: 01-June-2025 ]}