LeafCount

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`LeafCount`

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LeafCount[expr]

gives the total number of indivisible subexpressions in expr.

Details and Options

LeafCount gives a measure of the total "size" of an expression.
LeafCount counts the number of subexpressions in expr that correspond to "leaves" on the expression tree.
For ordinary expressions, LeafCount is based on FullForm representation.
For raw and other atomic expressions, the LeafCount is 1 with the following exceptions: »
Association based on the leaf count of the values

Complex based on FullForm

Rational based on FullForm
With the option setting HeadsFalse, LeafCount excludes heads of expressions and their parts.

Examples

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

Find the number of "leaves" in an expression tree:

In[1]:=1

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https://wolfram.com/xid/0en0qxga6-eg1

Out[1]=1

These are the leaves in the expression:

In[2]:=2

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https://wolfram.com/xid/0en0qxga6-t4r

Out[2]=2

Options (1)Common values & functionality for each option

Heads (1)

By default, leaves in heads are included:

In[1]:=1

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https://wolfram.com/xid/0en0qxga6-yyicyr

Out[1]=1

Exclude leaves in heads:

In[2]:=2

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https://wolfram.com/xid/0en0qxga6-8ir4m9

Out[2]=2

Applications (3)Sample problems that can be solved with this function

Find leaf count measures of the sizes of successive integrals:

In[1]:=1

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https://wolfram.com/xid/0en0qxga6-v9a

Out[1]=1

Make Simplify pick expressions with minimum LeafCount:

In[1]:=1

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https://wolfram.com/xid/0en0qxga6-rna

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0en0qxga6-ne2

Out[2]=2

Find leaf counts in combinator expressions [more info]:

In[1]:=1

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https://wolfram.com/xid/0en0qxga6-uog

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0en0qxga6-h2d

Out[2]=2

Properties & Relations (6)Properties of the function, and connections to other functions

LeafCount by default includes the head and all elements inside it:

In[3]:=3

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https://wolfram.com/xid/0en0qxga6-t63

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0en0qxga6-ms2

Out[4]=4

For expressions with atomic heads, this corresponds to the number of vertices in TreeForm[expr]:

In[5]:=5

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https://wolfram.com/xid/0en0qxga6-thr8bd

LeafCount[expr,HeadsFalse] typically gives the number of leaves in TreeForm[expr]:

In[1]:=1

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https://wolfram.com/xid/0en0qxga6-mq3s4p

In[2]:=2

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https://wolfram.com/xid/0en0qxga6-bah0hy

In[3]:=3

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https://wolfram.com/xid/0en0qxga6-6665sz

Out[3]=3

LeafCount counts elements in rational and complex numbers just as in FullForm:

In[1]:=1

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https://wolfram.com/xid/0en0qxga6-y0o

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0en0qxga6-tsl

LeafCount counts the head and values of an Association:

In[1]:=1

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https://wolfram.com/xid/0en0qxga6-m7ld0p

Out[1]=1

These are the five leaves in the expression:

In[2]:=2

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https://wolfram.com/xid/0en0qxga6-nx68ah

Out[2]=2

Reversing the keys and values gives a different result:

In[3]:=3

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https://wolfram.com/xid/0en0qxga6-lmjc8e

Out[3]=3

LeafCount treats atomic objects, such as SparseArray objects, as single expression leaves:

In[1]:=1

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https://wolfram.com/xid/0en0qxga6-tgc

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0en0qxga6-gpi

Out[2]=2

SparseArray objects have a complicated FullForm:

In[3]:=3

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https://wolfram.com/xid/0en0qxga6-kjiehv

But they are atomic:

In[4]:=4

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https://wolfram.com/xid/0en0qxga6-v3img8

Out[4]=4

For most expressions, LeafCount equals the count matching Blank[] at level {-1}:

In[1]:=1

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https://wolfram.com/xid/0en0qxga6-wyy4g

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0en0qxga6-jt63cw

Out[2]=2

LeafCount counts Rational and Complex numbers using their FullForm:

In[3]:=3

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https://wolfram.com/xid/0en0qxga6-14u8xv

In[4]:=4

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https://wolfram.com/xid/0en0qxga6-6c2vq

Out[4]=4

Count treats Rational and Complex as atoms:

In[5]:=5

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https://wolfram.com/xid/0en0qxga6-649t0c

Out[5]=5

In[6]:=6

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https://wolfram.com/xid/0en0qxga6-9q8uup

Out[6]=6

Possible Issues (3)Common pitfalls and unexpected behavior

LeafCount is based on the FullForm of expressions:

In[83]:=83

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https://wolfram.com/xid/0en0qxga6-nk

Out[83]=83

In[82]:=82

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https://wolfram.com/xid/0en0qxga6-egl

Out[82]=82

In[84]:=84

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https://wolfram.com/xid/0en0qxga6-cgk

LeafCount by default includes leaves in heads:

In[1]:=1

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https://wolfram.com/xid/0en0qxga6-2acen9

Out[1]=1

The head f is considered a leaf, even though it is not a leaf in a tree visualization of the expression:

In[2]:=2

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https://wolfram.com/xid/0en0qxga6-lx9jvu

Use the setting HeadsFalse to only count the leaves of the tree visualization:

In[3]:=3

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https://wolfram.com/xid/0en0qxga6-exb7zq

Out[3]=3

Unlike TreeForm, LeafCount takes into consideration the structure of expression heads:

In[1]:=1

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https://wolfram.com/xid/0en0qxga6-fgq0bo

In[2]:=2

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https://wolfram.com/xid/0en0qxga6-l16bv6

Out[2]=2

This shows the expression tree whose leaves are counted by LeafCount:

In[3]:=3

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https://wolfram.com/xid/0en0qxga6-dr0hkp

Out[3]=3

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