Pytrees

What is a pytree?

In JAX, a pytree is a container of leaf elements and/or more pytrees. Containers include lists, tuples, and dicts (JAX can be extended to consider other container types as pytrees, see Extending pytrees below). A leaf element is anything that’s not a pytree, e.g. an array. In other words, a pytree is just a possibly-nested standard or user-registered Python container. If nested, note that the container types do not need to match. A single “leaf”, i.e. a non-container object, is also considered a pytree.

Example pytrees:

[1, "a", object()] # 3 leaves

(1, (2, 3), ()) # 3 leaves

[1, {"k1": 2, "k2": (3, 4)}, 5] # 5 leaves

Pytrees and JAX functions

Many JAX functions, including all function transformations, operate over pytrees of arrays (other leaf types are sometimes allowed as well). Transformations are only applied to the leaf arrays while preserving the original pytree structure; for example, vmap and pmap only map over arrays, but automatically map over arrays inside of standard Python sequences, and can return mapped Python sequences.

Applying optional parameters to pytrees

Some JAX function transformations take optional parameters that specify how certain input or output values should be treated (e.g. the in_axes and out_axes arguments to vmap). These parameters are also pytrees, and the leaf values are “matched up” with the corresponding input or output leaf arrays. For example, if we pass the following input to vmap (note that the input arguments to a function are considered a tuple):

(a1, {"k1": a2, "k2": a3})

We can use the following in_axes pytree to specify that only the “k2” argument is mapped (axis=0) and the rest aren’t mapped over (axis=None):

(None, {"k1": None, "k2": 0})

Note that the optional parameter pytree structure must match that of the main input pytree. However, the optional parameters can optionally be specified as a “prefix” pytree, meaning that a single leaf value can be applied to an entire sub-pytree. For example, if we have the same vmap input as above, but wish to only map over the dictionary argument, we can use:

(None, 0)  # equivalent to (None, {"k1": 0, "k2": 0})

Or, if we want every argument to be mapped, we can simply write a single leaf value that is applied over the entire argument tuple pytree:

0

This happens to be the default in_axes value!

The same logic applies to other optional parameters that refer to specific input or output values of a transformed function, e.g. vmap’s out_axes and pmaps’s in_axes.

Developer information

This is primarily JAX internal documentation, end-users are not supposed to need to understand this to use JAX, except when registering new user-defined container types with JAX. Some of these details may change.

Internal pytree handling

JAX canonicalizes pytrees into flat lists of numeric or array types at the api.py boundary (and also in control flow primitives). This keeps downstream JAX internals simpler: vmap etc. can handle user functions that accept and return Python containers, while all the other parts of the system can operate on functions that only take (multiple) array arguments and always return a flat list of arrays.

When JAX flattens a pytree it will produce a list of leaves and a treedef object that encodes the structure of the original value. The treedef can then be used to construct a matching structured value after transforming the leaves. Pytrees are tree-like, rather than DAG-like or graph-like, in that we handle them assuming referential transparency and that they can’t contain reference cycles.

Here is a simple example:

from jax.tree_util import tree_flatten, tree_unflatten, register_pytree_node
import jax.numpy as jnp

# The structured value to be transformed
value_structured = [1., (2., 3.)]

# The leaves in value_flat correspond to the `*` markers in value_tree
value_flat, value_tree = tree_flatten(value_structured)
print("value_flat={}\nvalue_tree={}".format(value_flat, value_tree))

# Transform the flat value list using an element-wise numeric transformer
transformed_flat = list(map(lambda v: v * 2., value_flat))
print("transformed_flat={}".format(transformed_flat))

# Reconstruct the structured output, using the original
transformed_structured = tree_unflatten(value_tree, transformed_flat)
print("transformed_structured={}".format(transformed_structured))

# Output:
# value_flat=[1.0, 2.0, 3.0]
# value_tree=PyTreeDef(list, [*,PyTreeDef(tuple, [*,*])])
# transformed_flat=[2.0, 4.0, 6.0]
# transformed_structured=[2.0, (4.0, 6.0)]

By default, Pytrees containers can be lists, tuples, dicts, namedtuple, None, OrderedDict. Other types of values, including numeric and ndarray values, are treated as leaves:

from collections import namedtuple
Point = namedtuple('Point', ['x', 'y'])

example_containers = [
    (1., [2., 3.]),
    (1., {'b': 2., 'a': 3.}),
    1.,
    None,
    jnp.zeros(2),
    Point(1., 2.)
]
def show_example(structured):
  flat, tree = tree_flatten(structured)
  unflattened = tree_unflatten(tree, flat)
  print("structured={}\n  flat={}\n  tree={}\n  unflattened={}".format(
      structured, flat, tree, unflattened))

for structured in example_containers:
  show_example(structured)

# Output:
# structured=(1.0, [2.0, 3.0])
#   flat=[1.0, 2.0, 3.0]
#   tree=PyTreeDef(tuple, [*,PyTreeDef(list, [*,*])])
#   unflattened=(1.0, [2.0, 3.0])
# structured=(1.0, {'b': 2.0, 'a': 3.0})
#   flat=[1.0, 3.0, 2.0]
#   tree=PyTreeDef(tuple, [*,PyTreeDef(dict[['a', 'b']], [*,*])])
#   unflattened=(1.0, {'a': 3.0, 'b': 2.0})
# structured=1.0
#   flat=[1.0]
#   tree=*
#   unflattened=1.0
# structured=None
#   flat=[]
#   tree=PyTreeDef(None, [])
#   unflattened=None
# structured=[0. 0.]
#   flat=[DeviceArray([0., 0.], dtype=float32)]
#   tree=*
#   unflattened=[0. 0.]
# structured=Point(x=1.0, y=2.0)
#   flat=[1.0, 2.0]
#   tree=PyTreeDef(namedtuple[<class '__main__.Point'>], [*,*])
#   unflattened=Point(x=1.0, y=2.0)

Extending pytrees

By default, any part of a structured value that is not recognized as an internal pytree node is treated as a leaf (and such containers could not be passed to JAX-traceable functions):

class Special(object):
  def __init__(self, x, y):
    self.x = x
    self.y = y

  def __repr__(self):
    return "Special(x={}, y={})".format(self.x, self.y)


show_example(Special(1., 2.))

# Output:
# structured=Special(x=1.0, y=2.0)
#   flat=[Special(x=1.0, y=2.0)]
#   tree=*
#   unflattened=Special(x=1.0, y=2.0)

The set of Python types that are considered internal pytree nodes is extensible, through a global registry of types. Values of registered types are traversed recursively:

class RegisteredSpecial(Special):
def __repr__(self):
  return "RegisteredSpecial(x={}, y={})".format(self.x, self.y)

def special_flatten(v):
  """Specifies a flattening recipe.

  Params:
    v: the value of registered type to flatten.
  Returns:
    a pair of an iterable with the children to be flattened recursively,
    and some opaque auxiliary data to pass back to the unflattening recipe.
    The auxiliary data is stored in the treedef for use during unflattening.
    The auxiliary data could be used, e.g., for dictionary keys.
  """
  children = (v.x, v.y)
  aux_data = None
  return (children, aux_data)

def special_unflatten(aux_data, children):
  """Specifies an unflattening recipe.

  Params:
    aux_data: the opaque data that was specified during flattening of the
      current treedef.
    children: the unflattened children

  Returns:
    a re-constructed object of the registered type, using the specified
    children and auxiliary data.
  """
  return RegisteredSpecial(*children)

# Global registration
register_pytree_node(
    RegisteredSpecial,
    special_flatten,    # tell JAX what are the children nodes
    special_unflatten   # tell JAX how to pack back into a RegisteredSpecial
)

show_example(RegisteredSpecial(1., 2.))

# Output:
# structured=RegisteredSpecial(x=1.0, y=2.0)
#   flat=[1.0, 2.0]
#   tree=PyTreeDef(<class '__main__.RegisteredSpecial'>[None], [*,*])
#   unflattened=RegisteredSpecial(x=1.0, y=2.0)

JAX needs sometimes to compare treedef for equality. Therefore care must be taken to ensure that the auxiliary data specified in the flattening recipe supports a meaningful equality comparison.

The whole set of functions for operating on pytrees are in tree_util module.