bitarray 2.8.4

efficient arrays of booleans -- C extension

bitarray: efficient arrays of booleans

This library provides an object type which efficiently represents an array of booleans. Bitarrays are sequence types and behave very much like usual lists. Eight bits are represented by one byte in a contiguous block of memory. The user can select between two representations: little-endian and big-endian. All functionality is implemented in C. Methods for accessing the machine representation are provided, including the ability to import and export buffers. This allows creating bitarrays that are mapped to other objects, including memory-mapped files.

Roadmap

In 2024 (probably around July), we are planning the release of bitarray 3.0. The 3.0 release will:

• Remove Python 2.7 support.

• Make little the default bit-endianness, as it will make unaligned copying faster by default.

• Remove hackish support for bitarray() handling unpickling, see detailed explaination in #207. This will close #206.

Key features

• The bit endianness can be specified for each bitarray object, see below.

• Sequence methods: slicing (including slice assignment and deletion), operations +, *, +=, *=, the in operator, len()

• Bitwise operations: ~, &, |, ^, <<, >> (as well as their in-place versions &=, |=, ^=, <<=, >>=).

• Fast methods for encoding and decoding variable bit length prefix codes.

• Bitarray objects support the buffer protocol (both importing and exporting buffers).

• Packing and unpacking to other binary data formats, e.g. numpy.ndarray.

• Pickling and unpickling of bitarray objects.

• Immutable frozenbitarray objects which are hashable

• Sequential search

• Type hinting

• Extensive test suite with over 400 unittests.

• Utility module bitarray.util:

  • conversion to and from hexadecimal strings

  • (de-) serialization

  • pretty printing

  • conversion to and from integers

  • creating Huffman codes

  • compression of sparse bitarrays

  • various count functions

  • other helpful functions

Installation

Python wheels are are available on PyPI for all mayor platforms and Python versions. Which means you can simply:

$ pip install bitarray

In addition, conda packages are available (both the default Anaconda repository as well as conda-forge support bitarray):

$ conda install bitarray

Once you have installed the package, you may want to test it:

$ python -c 'import bitarray; bitarray.test()'
bitarray is installed in: /Users/ilan/bitarray/bitarray
bitarray version: 2.8.4
sys.version: 3.11.0 (main, Oct 25 2022) [Clang 14.0.4]
sys.prefix: /Users/ilan/Mini3/envs/py311
pointer size: 64 bit
sizeof(size_t): 8
sizeof(bitarrayobject): 80
__clang__ or __GNUC__ defined: 1
PY_LITTLE_ENDIAN (use word shift): 1
DEBUG: 0
.........................................................................
.........................................................................
................................................................
----------------------------------------------------------------------
Ran 494 tests in 0.461s

OK

The test() function is part of the API. It will return a unittest.runner.TextTestResult object, such that one can verify that all tests ran successfully by:

import bitarray
assert bitarray.test().wasSuccessful()

Usage

As mentioned above, bitarray objects behave very much like lists, so there is not too much to learn. The biggest difference from list objects (except that bitarray are obviously homogeneous) is the ability to access the machine representation of the object. When doing so, the bit endianness is of importance; this issue is explained in detail in the section below. Here, we demonstrate the basic usage of bitarray objects:

>>> from bitarray import bitarray
>>> a = bitarray()         # create empty bitarray
>>> a.append(1)
>>> a.extend([1, 0])
>>> a
bitarray('110')
>>> x = bitarray(2 ** 20)  # bitarray of length 1048576 (uninitialized)
>>> len(x)
1048576
>>> bitarray('1001 011')   # initialize from string (whitespace is ignored)
bitarray('1001011')
>>> lst = [1, 0, False, True, True]
>>> a = bitarray(lst)      # initialize from iterable
>>> a
bitarray('10011')
>>> a[2]    # indexing a single item will always return an integer
0
>>> a[2:4]  # whereas indexing a slice will always return a bitarray
bitarray('01')
>>> a[2:3]  # even when the slice length is just one
bitarray('0')
>>> a.count(1)
3
>>> a.remove(0)            # removes first occurrence of 0
>>> a
bitarray('1011')

Like lists, bitarray objects support slice assignment and deletion:

>>> a = bitarray(50)
>>> a.setall(0)            # set all elements in a to 0
>>> a[11:37:3] = 9 * bitarray('1')
>>> a
bitarray('00000000000100100100100100100100100100000000000000')
>>> del a[12::3]
>>> a
bitarray('0000000000010101010101010101000000000')
>>> a[-6:] = bitarray('10011')
>>> a
bitarray('000000000001010101010101010100010011')
>>> a += bitarray('000111')
>>> a[9:]
bitarray('001010101010101010100010011000111')

In addition, slices can be assigned to booleans, which is easier (and faster) than assigning to a bitarray in which all values are the same:

>>> a = 20 * bitarray('0')
>>> a[1:15:3] = True
>>> a
bitarray('01001001001001000000')

This is easier and faster than:

>>> a = 20 * bitarray('0')
>>> a[1:15:3] = 5 * bitarray('1')
>>> a
bitarray('01001001001001000000')

Note that in the latter we have to create a temporary bitarray whose length must be known or calculated. Another example of assigning slices to Booleans, is setting ranges:

>>> a = bitarray(30)
>>> a[:] = 0         # set all elements to 0 - equivalent to a.setall(0)
>>> a[10:25] = 1     # set elements in range(10, 25) to 1
>>> a
bitarray('000000000011111111111111100000')

As of bitarray version 2.8, indices may also be lists of arbitrary indices (like in NumPy), or bitarrays that are treated as masks, see Bitarray indexing.

Bitwise operators

Bitarray objects support the bitwise operators ~, &, |, ^, <<, >> (as well as their in-place versions &=, |=, ^=, <<=, >>=). The behavior is very much what one would expect:

>>> a = bitarray('101110001')
>>> ~a  # invert
bitarray('010001110')
>>> b = bitarray('111001011')
>>> a ^ b
bitarray('010111010')
>>> a &= b
>>> a
bitarray('101000001')
>>> a <<= 2   # in-place left shift by 2
>>> a
bitarray('100000100')
>>> b >> 1
bitarray('011100101')

The C language does not specify the behavior of negative shifts and of left shifts larger or equal than the width of the promoted left operand. The exact behavior is compiler/machine specific. This Python bitarray library specifies the behavior as follows:

• the length of the bitarray is never changed by any shift operation

• blanks are filled by 0

• negative shifts raise ValueError

• shifts larger or equal to the length of the bitarray result in bitarrays with all values 0

It is worth noting that (regardless of bit endianness) the bitarray left shift (<<) always shifts towards lower indices, and the right shift (>>) always shifts towards higher indices.

Bit endianness

Unless explicitly converting to machine representation, using the .tobytes(), .frombytes(), .tofile() and .fromfile() methods, as well as using memoryview, the bit endianness will have no effect on any computation, and one can skip this section.

Since bitarrays allows addressing individual bits, where the machine represents 8 bits in one byte, there are two obvious choices for this mapping: little-endian and big-endian.

When dealing with the machine representation of bitarray objects, it is recommended to always explicitly specify the endianness.

By default, bitarrays use big-endian representation:

>>> a = bitarray()
>>> a.endian()
'big'
>>> a.frombytes(b'A')
>>> a
bitarray('01000001')
>>> a[6] = 1
>>> a.tobytes()
b'C'

Big-endian means that the most-significant bit comes first. Here, a[0] is the lowest address (index) and most significant bit, and a[7] is the highest address and least significant bit.

When creating a new bitarray object, the endianness can always be specified explicitly:

>>> a = bitarray(endian='little')
>>> a.frombytes(b'A')
>>> a
bitarray('10000010')
>>> a.endian()
'little'

Here, the low-bit comes first because little-endian means that increasing numeric significance corresponds to an increasing address. So a[0] is the lowest address and least significant bit, and a[7] is the highest address and most significant bit.

The bit endianness is a property of the bitarray object. The endianness cannot be changed once a bitarray object is created. When comparing bitarray objects, the endianness (and hence the machine representation) is irrelevant; what matters is the mapping from indices to bits:

>>> bitarray('11001', endian='big') == bitarray('11001', endian='little')
True

Bitwise operations (|, ^, &=, |=, ^=, ~) are implemented efficiently using the corresponding byte operations in C, i.e. the operators act on the machine representation of the bitarray objects. Therefore, it is not possible to perform bitwise operators on bitarrays with different endianness.

When converting to and from machine representation, using the .tobytes(), .frombytes(), .tofile() and .fromfile() methods, the endianness matters:

>>> a = bitarray(endian='little')
>>> a.frombytes(b'\x01')
>>> a
bitarray('10000000')
>>> b = bitarray(endian='big')
>>> b.frombytes(b'\x80')
>>> b
bitarray('10000000')
>>> a == b
True
>>> a.tobytes() == b.tobytes()
False

As mentioned above, the endianness can not be changed once an object is created. However, you can create a new bitarray with different endianness:

>>> a = bitarray('111000', endian='little')
>>> b = bitarray(a, endian='big')
>>> b
bitarray('111000')
>>> a == b
True

Buffer protocol

Bitarray objects support the buffer protocol. They can both export their own buffer, as well as import another object’s buffer. To learn more about this topic, please read buffer protocol. There is also an example that shows how to memory-map a file to a bitarray: mmapped-file.py

Variable bit length prefix codes

The .encode() method takes a dictionary mapping symbols to bitarrays and an iterable, and extends the bitarray object with the encoded symbols found while iterating. For example:

>>> d = {'H':bitarray('111'), 'e':bitarray('0'),
...      'l':bitarray('110'), 'o':bitarray('10')}
...
>>> a = bitarray()
>>> a.encode(d, 'Hello')
>>> a
bitarray('111011011010')

Note that the string 'Hello' is an iterable, but the symbols are not limited to characters, in fact any immutable Python object can be a symbol. Taking the same dictionary, we can apply the .decode() method which will return a list of the symbols:

>>> a.decode(d)
['H', 'e', 'l', 'l', 'o']
>>> ''.join(a.decode(d))
'Hello'

Since symbols are not limited to being characters, it is necessary to return them as elements of a list, rather than simply returning the joined string. The above dictionary d can be efficiently constructed using the function bitarray.util.huffman_code(). I also wrote Huffman coding in Python using bitarray for more background information.

When the codes are large, and you have many decode calls, most time will be spent creating the (same) internal decode tree objects. In this case, it will be much faster to create a decodetree object, which can be passed to bitarray’s .decode() and .iterdecode() methods, instead of passing the prefix code dictionary to those methods itself:

>>> from bitarray import bitarray, decodetree
>>> t = decodetree({'a': bitarray('0'), 'b': bitarray('1')})
>>> a = bitarray('0110')
>>> a.decode(t)
['a', 'b', 'b', 'a']
>>> ''.join(a.iterdecode(t))
'abba'

The sole purpose of the immutable decodetree object is to be passed to bitarray’s .decode() and .iterdecode() methods.

Frozenbitarrays

A frozenbitarray object is very similar to the bitarray object. The difference is that this a frozenbitarray is immutable, and hashable, and can therefore be used as a dictionary key:

>>> from bitarray import frozenbitarray
>>> key = frozenbitarray('1100011')
>>> {key: 'some value'}
{frozenbitarray('1100011'): 'some value'}
>>> key[3] = 1
Traceback (most recent call last):
    ...
TypeError: frozenbitarray is immutable

Reference

bitarray version: 2.8.4 – change log

In the following, item and value are usually a single bit - an integer 0 or 1.

The bitarray object:

bitarray(initializer=0, /, endian='big', buffer=None) -> bitarray

Return a new bitarray object whose items are bits initialized from the optional initial object, and endianness. The initializer may be of the following types:

int: Create a bitarray of given integer length. The initial values are uninitialized.

str: Create bitarray from a string of 0 and 1.

iterable: Create bitarray from iterable or sequence of integers 0 or 1.

Optional keyword arguments:

endian: Specifies the bit endianness of the created bitarray object. Allowed values are big and little (the default is big). The bit endianness effects the buffer representation of the bitarray.

buffer: Any object which exposes a buffer. When provided, initializer cannot be present (or has to be None). The imported buffer may be read-only or writable, depending on the object type.

New in version 2.3: optional buffer argument.

bitarray methods:

all() -> bool

Return True when all bits in bitarray are True. Note that a.all() is faster than all(a).

any() -> bool

Return True when any bit in bitarray is True. Note that a.any() is faster than any(a).

append(item, /)

Append item to the end of the bitarray.

buffer_info() -> tuple

Return a tuple containing:

0. memory address of buffer

1. buffer size (in bytes)

2. bit endianness as a string

3. number of pad bits

4. allocated memory for the buffer (in bytes)

5. memory is read-only

6. buffer is imported

7. number of buffer exports

bytereverse(start=0, stop=<end of buffer>, /)

For each byte in byte-range(start, stop) reverse the bit order in-place. The start and stop indices are given in terms of bytes (not bits). Also note that this method only changes the buffer; it does not change the endianness of the bitarray object. Padbits are left unchanged such that two consecutive calls will always leave the bitarray unchanged.

New in version 2.2.5: optional start and stop arguments.

clear()

Remove all items from the bitarray.

New in version 1.4.

copy() -> bitarray

Return a copy of the bitarray.

count(value=1, start=0, stop=<end of array>, step=1, /) -> int

Count number of occurrences of value in the bitarray.

New in version 1.1.0: optional start and stop arguments.

New in version 2.3.7: optional step argument.

decode(code, /) -> list

Given a prefix code (a dict mapping symbols to bitarrays, or decodetree object), decode content of bitarray and return it as a list of symbols.

encode(code, iterable, /)

Given a prefix code (a dict mapping symbols to bitarrays), iterate over the iterable object with symbols, and extend bitarray with corresponding bitarray for each symbol.

endian() -> str

Return the bit endianness of the bitarray as a string (little or big).

extend(iterable, /)

Append all items from iterable to the end of the bitarray. If the iterable is a string, each 0 and 1 are appended as bits (ignoring whitespace and underscore).

fill() -> int

Add zeros to the end of the bitarray, such that the length will be a multiple of 8, and return the number of bits added (0..7).

find(sub_bitarray, start=0, stop=<end of array>, /) -> int

Return lowest index where sub_bitarray is found, such that sub_bitarray is contained within [start:stop]. Return -1 when sub_bitarray is not found.

New in version 2.1.

frombytes(bytes, /)

Extend bitarray with raw bytes from a bytes-like object. Each added byte will add eight bits to the bitarray.

New in version 2.5.0: allow bytes-like argument.

fromfile(f, n=-1, /)

Extend bitarray with up to n bytes read from file object f (or any other binary stream what supports a .read() method, e.g. io.BytesIO). Each read byte will add eight bits to the bitarray. When n is omitted or negative, all bytes until EOF are read. When n is non-negative but exceeds the data available, EOFError is raised (but the available data is still read and appended).

index(sub_bitarray, start=0, stop=<end of array>, /) -> int

Return lowest index where sub_bitarray is found, such that sub_bitarray is contained within [start:stop]. Raises ValueError when the sub_bitarray is not present.

insert(index, value, /)

Insert value into bitarray before index.

invert(index=<all bits>, /)

Invert all bits in bitarray (in-place). When the optional index is given, only invert the single bit at index.

New in version 1.5.3: optional index argument.

iterdecode(code, /) -> iterator

Given a prefix code (a dict mapping symbols to bitarrays, or decodetree object), decode content of bitarray and return an iterator over the symbols.

itersearch(sub_bitarray, /) -> iterator

Searches for given sub_bitarray in self, and return an iterator over the start positions where sub_bitarray matches self.

pack(bytes, /)

Extend bitarray from a bytes-like object, where each byte corresponds to a single bit. The byte b'\x00' maps to bit 0 and all other bytes map to bit 1.

This method, as well as the .unpack() method, are meant for efficient transfer of data between bitarray objects to other Python objects (for example NumPy’s ndarray object) which have a different memory view.

New in version 2.5.0: allow bytes-like argument.

pop(index=-1, /) -> item

Remove and return item at index (default last). Raises IndexError if index is out of range.

remove(value, /)

Remove the first occurrence of value. Raises ValueError if value is not present.

reverse()

Reverse all bits in bitarray (in-place).

search(sub_bitarray, limit=<none>, /) -> list

Searches for given sub_bitarray in self, and return list of start positions. The optional argument limits the number of search results to the integer specified. By default, all search results are returned.

setall(value, /)

Set all elements in bitarray to value. Note that a.setall(value) is equivalent to a[:] = value.

sort(reverse=False)

Sort all bits in bitarray (in-place).

to01() -> str

Return a string containing ‘0’s and ‘1’s, representing the bits in the bitarray.

tobytes() -> bytes

Return the bitarray buffer in bytes (pad bits are set to zero).

tofile(f, /)

Write byte representation of bitarray to file object f.

tolist() -> list

Return bitarray as list of integer items. a.tolist() is equal to list(a).

Note that the list object being created will require 32 or 64 times more memory (depending on the machine architecture) than the bitarray object, which may cause a memory error if the bitarray is very large.

unpack(zero=b'\x00', one=b'\x01') -> bytes

Return bytes containing one character for each bit in the bitarray, using specified mapping.

bitarray data descriptors:

Data descriptors were added in version 2.6.

nbytes -> int

buffer size in bytes

padbits -> int

number of pad bits

readonly -> bool

bool indicating whether buffer is read-only

Other objects:

frozenbitarray(initializer=0, /, endian='big', buffer=None) -> frozenbitarray

Return a frozenbitarray object. Initialized the same way a bitarray object is initialized. A frozenbitarray is immutable and hashable, and may therefore be used as a dictionary key.

New in version 1.1.

decodetree(code, /) -> decodetree

Given a prefix code (a dict mapping symbols to bitarrays), create a binary tree object to be passed to .decode() or .iterdecode().

New in version 1.6.

Functions defined in the bitarray module:

bits2bytes(n, /) -> int

Return the number of bytes necessary to store n bits.

get_default_endian() -> str

Return the default endianness for new bitarray objects being created. Unless _set_default_endian('little') was called, the default endianness is big.

New in version 1.3.

test(verbosity=1) -> TextTestResult

Run self-test, and return unittest.runner.TextTestResult object.

Functions defined in bitarray.util module:

This sub-module was added in version 1.2.

zeros(length, /, endian=None) -> bitarray

Create a bitarray of length, with all values 0, and optional endianness, which may be ‘big’, ‘little’.

urandom(length, /, endian=None) -> bitarray

Return a bitarray of length random bits (uses os.urandom).

New in version 1.7.

pprint(bitarray, /, stream=None, group=8, indent=4, width=80)

Prints the formatted representation of object on stream (which defaults to sys.stdout). By default, elements are grouped in bytes (8 elements), and 8 bytes (64 elements) per line. Non-bitarray objects are printed by the standard library function pprint.pprint().

New in version 1.8.

make_endian(bitarray, /, endian) -> bitarray

When the endianness of the given bitarray is different from endian, return a new bitarray, with endianness endian and the same elements as the original bitarray. Otherwise (endianness is already endian) the original bitarray is returned unchanged.

New in version 1.3.

rindex(bitarray, value=1, start=0, stop=<end of array>, /) -> int

Return rightmost (highest) index of value in bitarray. Raises ValueError if value is not present.

New in version 2.3.0: optional start and stop arguments.

strip(bitarray, /, mode='right') -> bitarray

Return a new bitarray with zeros stripped from left, right or both ends. Allowed values for mode are the strings: left, right, both

count_n(a, n, value=1, /) -> int

Return lowest index i for which a[:i].count(value) == n. Raises ValueError when n exceeds total count (a.count(value)).

New in version 2.3.6: optional value argument.

parity(a, /) -> int

Return parity of bitarray a. parity(a) is equivalent to a.count() % 2 but more efficient.

New in version 1.9.

count_and(a, b, /) -> int

Return (a & b).count() in a memory efficient manner, as no intermediate bitarray object gets created.

count_or(a, b, /) -> int

Return (a | b).count() in a memory efficient manner, as no intermediate bitarray object gets created.

count_xor(a, b, /) -> int

Return (a ^ b).count() in a memory efficient manner, as no intermediate bitarray object gets created.

This is also known as the Hamming distance.

any_and(a, b, /) -> bool

Efficient implementation of any(a & b).

New in version 2.7.

subset(a, b, /) -> bool

Return True if bitarray a is a subset of bitarray b. subset(a, b) is equivalent to a | b == b (and equally a & b == a) but more efficient as no intermediate bitarray object is created and the buffer iteration is stopped as soon as one mismatch is found.

intervals(bitarray, /) -> iterator

Compute all uninterrupted intervals of 1s and 0s, and return an iterator over tuples (value, start, stop). The intervals are guaranteed to be in order, and their size is always non-zero (stop - start > 0).

New in version 2.7.

ba2hex(bitarray, /) -> hexstr

Return a string containing the hexadecimal representation of the bitarray (which has to be multiple of 4 in length).

hex2ba(hexstr, /, endian=None) -> bitarray

Bitarray of hexadecimal representation. hexstr may contain any number (including odd numbers) of hex digits (upper or lower case).

ba2base(n, bitarray, /) -> str

Return a string containing the base n ASCII representation of the bitarray. Allowed values for n are 2, 4, 8, 16, 32 and 64. The bitarray has to be multiple of length 1, 2, 3, 4, 5 or 6 respectively. For n=16 (hexadecimal), ba2hex() will be much faster, as ba2base() does not take advantage of byte level operations. For n=32 the RFC 4648 Base32 alphabet is used, and for n=64 the standard base 64 alphabet is used.

See also: Bitarray representations

New in version 1.9.

base2ba(n, asciistr, /, endian=None) -> bitarray

Bitarray of base n ASCII representation. Allowed values for n are 2, 4, 8, 16, 32 and 64. For n=16 (hexadecimal), hex2ba() will be much faster, as base2ba() does not take advantage of byte level operations. For n=32 the RFC 4648 Base32 alphabet is used, and for n=64 the standard base 64 alphabet is used.

See also: Bitarray representations

New in version 1.9.

ba2int(bitarray, /, signed=False) -> int

Convert the given bitarray to an integer. The bit-endianness of the bitarray is respected. signed indicates whether two’s complement is used to represent the integer.

int2ba(int, /, length=None, endian=None, signed=False) -> bitarray

Convert the given integer to a bitarray (with given endianness, and no leading (big-endian) / trailing (little-endian) zeros), unless the length of the bitarray is provided. An OverflowError is raised if the integer is not representable with the given number of bits. signed determines whether two’s complement is used to represent the integer, and requires length to be provided.

serialize(bitarray, /) -> bytes

Return a serialized representation of the bitarray, which may be passed to deserialize(). It efficiently represents the bitarray object (including its bit-endianness) and is guaranteed not to change in future releases.

See also: Bitarray representations

New in version 1.8.

deserialize(bytes, /) -> bitarray

Return a bitarray given a bytes-like representation such as returned by serialize().

See also: Bitarray representations

New in version 1.8.

New in version 2.5.0: allow bytes-like argument.

sc_encode(bitarray, /) -> bytes

Compress a sparse bitarray and return its binary representation. This representation is useful for efficiently storing sparse bitarrays. Use sc_decode() for decompressing (decoding).

See also: Compression of sparse bitarrays

New in version 2.7.

sc_decode(stream) -> bitarray

Decompress binary stream (an integer iterator, or bytes-like object) of a sparse compressed (sc) bitarray, and return the decoded bitarray. This function consumes only one bitarray and leaves the remaining stream untouched. Use sc_encode() for compressing (encoding).

See also: Compression of sparse bitarrays

New in version 2.7.

vl_encode(bitarray, /) -> bytes

Return variable length binary representation of bitarray. This representation is useful for efficiently storing small bitarray in a binary stream. Use vl_decode() for decoding.

See also: Variable length bitarray format

New in version 2.2.

vl_decode(stream, /, endian=None) -> bitarray

Decode binary stream (an integer iterator, or bytes-like object), and return the decoded bitarray. This function consumes only one bitarray and leaves the remaining stream untouched. Use vl_encode() for encoding.

See also: Variable length bitarray format

New in version 2.2.

huffman_code(dict, /, endian=None) -> dict

Given a frequency map, a dictionary mapping symbols to their frequency, calculate the Huffman code, i.e. a dict mapping those symbols to bitarrays (with given endianness). Note that the symbols are not limited to being strings. Symbols may may be any hashable object (such as None).

canonical_huffman(dict, /) -> tuple

Given a frequency map, a dictionary mapping symbols to their frequency, calculate the canonical Huffman code. Returns a tuple containing:

0. the canonical Huffman code as a dict mapping symbols to bitarrays

1. a list containing the number of symbols of each code length

2. a list of symbols in canonical order

Note: the two lists may be used as input for canonical_decode().

See also: Canonical Huffman Coding

New in version 2.5.

canonical_decode(bitarray, count, symbol, /) -> iterator

Decode bitarray using canonical Huffman decoding tables where count is a sequence containing the number of symbols of each length and symbol is a sequence of symbols in canonical order.

See also: Canonical Huffman Coding

New in version 2.5.