optree 0.4.0

Optimized PyTree Utilities.

OpTree

Optimized PyTree Utilities.

---

Table of Contents

• Installation
• PyTrees
  • Tree Nodes and Leaves
    • Built-in PyTree Node Types
    • Registering a Custom Container-like Type as Non-leaf Nodes
    • Notes about the PyTree Type Registry
  • None is non-leaf Node vs. None is Leaf
  • Key Ordering for Dictionaries
• Benchmark
  • Tree Flatten
  • Tree UnFlatten
  • Tree Copy
  • Tree Map
  • Tree Map (nargs)
• License

---

Installation

Install from PyPI ( / ):

pip3 install --upgrade optree

Install from conda-forge ():

conda install -c conda-forge optree

Install the latest version from GitHub:

pip3 install git+https://github.com/metaopt/optree.git#egg=optree

Or, clone this repo and install manually:

git clone --depth=1 https://github.com/metaopt/optree.git
cd optree
pip3 install .

Compiling from the source requires Python 3.7+, a compiler (gcc / clang / icc / cl.exe) supports C++20 and a cmake installation.

---

PyTrees

A PyTree is a recursive structure that can be an arbitrarily nested Python container (e.g., tuple, list, dict, OrderedDict, NamedTuple, etc.) or an opaque Python object. The key concepts of tree operations are tree flattening and its inverse (tree unflattening). Additional tree operations can be performed based on these two basic functions (e.g., tree_map = tree_unflatten ∘ map ∘ tree_flatten).

Tree flattening is traversing the entire tree in a left-to-right depth-first manner and returning the leaves of the tree in a deterministic order.

>>> tree = {'b': (2, [3, 4]), 'a': 1, 'c': 5, 'd': 6}
>>> optree.tree_flatten(tree)
([1, 2, 3, 4, 5, 6], PyTreeSpec({'a': *, 'b': (*, [*, *]), 'c': *, 'd': *}))
>>> optree.tree_flatten(1)
([1], PyTreeSpec(*))
>>> optree.tree_flatten(None)
([], PyTreeSpec(None))

This usually implies that the equal pytrees return equal lists of leaves and the same tree structure. See also section Key Ordering for Dictionaries.

>>> {'a': [1, 2], 'b': [3]} == {'b': [3], 'a': [1, 2]}
True
>>> optree.tree_leaves({'a': [1, 2], 'b': [3]}) == optree.tree_leaves({'b': [3], 'a': [1, 2]})
True
>>> optree.tree_structure({'a': [1, 2], 'b': [3]}) == optree.tree_structure({'b': [3], 'a': [1, 2]})
True

Tree Nodes and Leaves

A tree is a collection of non-leaf nodes and leaf nodes, where the leaf nodes have no children to flatten. optree.tree_flatten(...) will flatten the tree and return a list of leaf nodes while the non-leaf nodes will store in the tree specification.

Built-in PyTree Node Types

OpTree out-of-box supports the following Python container types in the registry:

• tuple
• list
• dict
• collections.namedtuple and its subclasses
• collections.OrderedDict
• collections.defaultdict
• collections.deque

which are considered non-leaf nodes in the tree. Python objects that the type is not registered will be treated as leaf nodes. The registration lookup uses the is operator to determine whether the type is matched. So subclasses will need to explicitly register in the registration, otherwise, an object of that type will be considered as a leaf. The NoneType is a special case discussed in section None is non-leaf Node vs. None is Leaf.

Registering a Custom Container-like Type as Non-leaf Nodes

A container-like Python type can be registered in the container registry with a pair of functions that specify:

• flatten_func(container) -> (children, metadata, entries): convert an instance of the container type to a (children, metadata, entries) triple, where children is an iterable of subtrees and entries is an iterable of path entries of the container (e.g., indices or keys).
• unflatten_func(metadata, children) -> container: convert such a pair back to an instance of the container type.

The metadata is some necessary data apart from the children to reconstruct the container, e.g., the keys of the dictionary (the children are values).

The entries can be omitted (only returns a pair) or is optional to implement (returns None). If so, use range(len(children)) (i.e., flat indices) as path entries of the current node. The function signature can be flatten_func(container) -> (children, metadata) or flatten_func(container) -> (children, metadata, None).

# Registry a Python type with lambda functions
>>> register_pytree_node(
...     set,
...     # (set) -> (children, metadata, None)
...     lambda s: (sorted(s), None, None),
...     # (metadata, children) -> (set)
...     lambda _, children: set(children),
...     namespace='set',
... )

# Register a Python type into a namespace
>>> import torch
>>> register_pytree_node(
...     torch.Tensor,
...     # (tensor) -> (children, metadata)
...     flatten_func=lambda tensor: (
...         (tensor.cpu().numpy(),),
...         dict(dtype=tensor.dtype, device=tensor.device, requires_grad=tensor.requires_grad),
...     ),
...     # (metadata, children) -> tensor
...     unflatten_func=lambda metadata, children: torch.tensor(children[0], **metadata),
...     namespace='torch2numpy',
... )
<class 'torch.Tensor'>

>>> tree = {'weight': torch.ones(size=(1, 2)).cuda(), 'bias': torch.zeros(size=(2,))}
>>> tree
{'weight': tensor([[1., 1.]], device='cuda:0'), 'bias': tensor([0., 0.])}

# Flatten without specifying the namespace
>>> tree_flatten(tree)  # `torch.Tensor`s are leaf nodes
([tensor([0., 0.]), tensor([[1., 1.]], device='cuda:0')], PyTreeSpec({'bias': *, 'weight': *}))

# Flatten with the namespace
>>> leaves, treespec = optree.tree_flatten(tree, namespace='torch2numpy')
>>> leaves, treespec
(
    [array([0., 0.], dtype=float32), array([[1., 1.]], dtype=float32)],
    PyTreeSpec(
        {
            'bias': CustomTreeNode(Tensor[{'dtype': torch.float32, 'device': device(type='cpu'), 'requires_grad': False}], [*]),
            'weight': CustomTreeNode(Tensor[{'dtype': torch.float32, 'device': device(type='cuda', index=0), 'requires_grad': False}], [*])
        },
        namespace='torch2numpy'
    )
)

# `entries` are not defined and use `range(len(children))`
>>> optree.tree_paths(tree, namespace='torch2numpy')
[('bias', 0), ('weight', 0)]

# Unflatten back to a copy of the original object
>>> optree.tree_unflatten(treespec, leaves)
{'bias': tensor([0., 0.]), 'weight': tensor([[1., 1.]], device='cuda:0')}

Users can also extend the pytree registry by decorating the custom class and defining an instance method tree_flatten and a class method tree_unflatten.

>>> from collections import UserDict
...
... @optree.register_pytree_node_class(namespace='mydict')
... class MyDict(UserDict):
...     def tree_flatten(self):  # -> (children, metadata, entries)
...         reversed_keys = sorted(self.keys(), reverse=True)
...         return (
...             [self[key] for key in reversed_keys],  # children
...             reversed_keys,  # metadata
...             reversed_keys,  # entries
...         )
...
...     @classmethod
...     def tree_unflatten(cls, metadata, children):
...         return cls(zip(metadata, children))

>>> tree = MyDict(b=4, a=(2, 3), c=MyDict({'d': 5, 'f': 6}))

# Flatten without specifying the namespace
>>> optree.tree_flatten_with_path(tree)  # `MyDict`s are leaf nodes
(
    [()],
    [MyDict(b=4, a=(2, 3), c=MyDict({'d': 5, 'f': 6}))],
    PyTreeSpec(*)
)

# Flatten with the namespace
>>> optree.tree_flatten_with_path(tree, namespace='mydict')
(
    [('c', 'f'), ('c', 'd'), ('b',), ('a', 0), ('a', 1)],
    [6, 5, 4, 2, 3],
    PyTreeSpec(
        CustomTreeNode(MyDict[['c', 'b', 'a']], [CustomTreeNode(MyDict[['f', 'd']], [*, *]), *, (*, *)]),
        namespace='mydict'
    )
)

Notes about the PyTree Type Registry

There are several key attributes of the pytree type registry:

1. The type registry is per-interpreter-dependent. This means registering a custom type in the registry affects all modules that use OpTree.

   - !!! WARNING !!!
     For safety reasons, a `namespace` must be specified while registering a custom type. It is
     used to isolate the behavior of flattening and unflattening a pytree node type. This is to
     prevent accidental collisions between different libraries that may register the same type.
   

2. The elements in the type registry are immutable. Users either cannot register the same type twice in the same namespace (i.e., update the type registry). Nor cannot remove a type from the type registry. To update the behavior of an already registered type, simply register it again with another namespace.

3. Users cannot modify the behavior of already registered built-in types listed Built-in PyTree Node Types, such as key order sorting for dict and collections.defaultdict.

4. Inherited subclasses are not implicitly registered. The registration lookup uses type(obj) is registered_type rather than isinstance(obj, registered_type). Users need to register the subclasses explicitly. To register all subclasses, it is easy to implement with metaclass or __init_subclass__, for example:

   from collections import UserDict
   
   @optree.register_pytree_node_class(namespace='mydict')
   class MyDict(UserDict):
       def __init_subclass__(cls):  # define this in the base class
           super().__init_subclass__()
           # Register a subclass to namespace 'mydict'
           optree.register_pytree_node_class(cls, namespace='mydict')
   
       def tree_flatten(self):  # -> (children, metadata, entries)
           reversed_keys = sorted(self.keys(), reverse=True)
           return (
               [self[key] for key in reversed_keys],  # children
               reversed_keys,  # metadata
               reversed_keys,  # entries
           )
   
       @classmethod
       def tree_unflatten(cls, metadata, children):
           return cls(zip(metadata, children))
   
   # Subclasses will be automatically registered in namespace 'mydict'
   class MyAnotherDict(MyDict):
       pass
   
   tree = MyDict(b=4, a=(2, 3), c=MyAnotherDict({'d': 5, 'f': 6}))
   optree.tree_flatten_with_path(tree, namespace='mydict')
   # (
   #     [('c', 'f'), ('c', 'd'), ('b',), ('a', 0), ('a', 1)],
   #     [6, 5, 4, 2, 3],
   #     PyTreeSpec(CustomTreeNode(MyDict[['c', 'b', 'a']], [CustomTreeNode(MyAnotherDict[['f', 'd']], [*, *]), *, (*, *)]), namespace='mydict')
   # )
   

None is non-leaf Node vs. None is Leaf

The None object is a special object in the Python language. It serves some of the same purposes as null (a pointer does not point to anything) in other programming languages, which denotes a variable is empty or marks default parameters. However, the None object is a singleton object rather than a pointer. It may also serve as a sentinel value. In addition, if a function has returned without any return value, it also implicitly returns the None object.

By default, the None object is considered a non-leaf node in the tree with arity 0, i.e., a non-leaf node that has no children. This is like the behavior of an empty tuple. While flattening a tree, it will remain in the tree structure definitions rather than in the leaves list.

>>> tree = {'b': (2, [3, 4]), 'a': 1, 'c': None, 'd': 5}
>>> optree.tree_flatten(tree)
([1, 2, 3, 4, 5], PyTreeSpec({'a': *, 'b': (*, [*, *]), 'c': None, 'd': *}))
>>> optree.tree_flatten(tree, none_is_leaf=True)
([1, 2, 3, 4, None, 5], PyTreeSpec({'a': *, 'b': (*, [*, *]), 'c': *, 'd': *}, NoneIsLeaf))
>>> optree.tree_flatten(1)
([1], PyTreeSpec(*))
>>> optree.tree_flatten(None)
([], PyTreeSpec(None))
>>> optree.tree_flatten(None, none_is_leaf=True)
([None], PyTreeSpec(*, NoneIsLeaf))

OpTree provides a keyword argument none_is_leaf to determine whether to consider the None object as a leaf, like other opaque objects. If none_is_leaf=True, the None object will place in the leaves list. Otherwise, the None object will remain in the tree specification (structure).

>>> import torch

>>> linear = torch.nn.Linear(in_features=3, out_features=2, bias=False)
>>> linear._parameters
OrderedDict([
    ('weight', Parameter containing:
               tensor([[-0.6677,  0.5209,  0.3295],
                       [-0.4876, -0.3142,  0.1785]], requires_grad=True)),
    ('bias', None)
])

>>> optree.tree_map(torch.zeros_like, linear._parameters)
OrderedDict([
    ('weight', tensor([[0., 0., 0.],
                       [0., 0., 0.]])),
    ('bias', None)
])

>>> optree.tree_map(torch.zeros_like, linear._parameters, none_is_leaf=True)
TypeError: zeros_like(): argument 'input' (position 1) must be Tensor, not NoneType
>>> optree.tree_map(lambda t: torch.zeros_like(t) if t is not None else 0, linear._parameters, none_is_leaf=True)
OrderedDict([
    ('weight', tensor([[0., 0., 0.],
                       [0., 0., 0.]])),
    ('bias', 0)
])

Key Ordering for Dictionaries

The built-in Python dictionary (i.e., builtins.dict) is an unordered mapping that holds the keys and values. The leaves of a dictionary are the values. Although since Python 3.6, the built-in dictionary is insertion ordered (PEP 468). The dictionary equality operator (==) does not check for key ordering. To ensure that "equal dict" implies "equal ordering of leaves", the order of values of the dictionary is sorted by the keys. This behavior is also applied to collections.defaultdict.

>>> optree.tree_flatten({'a': [1, 2], 'b': [3]})
([1, 2, 3], PyTreeSpec({'a': [*, *], 'b': [*]}))
>>> optree.tree_flatten({'b': [3], 'a': [1, 2]})
([1, 2, 3], PyTreeSpec({'a': [*, *], 'b': [*]}))

Note that there are no restrictions on the dict to require the keys are comparable (sortable). There can be multiple types of keys in the dictionary. The keys are sorted in ascending order by key=lambda k: k first if capable otherwise fallback to key=lambda k: (k.__class__.__qualname__, k). This handles most cases.

>>> sorted({1: 2, 1.5: 1}.keys())
[1, 1.5]
>>> sorted({'a': 3, 1: 2, 1.5: 1}.keys())
TypeError: '<' not supported between instances of 'int' and 'str'
>>> sorted({'a': 3, 1: 2, 1.5: 1}.keys(), key=lambda k: (k.__class__.__qualname__, k))
[1.5, 1, 'a']

If users want to keep the values in the insertion order, they should use collection.OrderedDict, which will take the order of keys under consideration:

>>> OrderedDict([('a', [1, 2]), ('b', [3])]) == OrderedDict([('b', [3]), ('a', [1, 2])])
False
>>> optree.tree_flatten(OrderedDict([('a', [1, 2]), ('b', [3])]))
([1, 2, 3], PyTreeSpec(OrderedDict([('a', [*, *]), ('b', [*])])))
>>> optree.tree_flatten(OrderedDict([('b', [3]), ('a', [1, 2])]))
([3, 1, 2], PyTreeSpec(OrderedDict([('b', [*]), ('a', [*, *])])))

---

Benchmark

We benchmark the performance of:

• tree flatten
• tree unflatten
• tree copy (i.e., unflatten(flatten(...)))
• tree map

compared with the following libraries:

• OpTree (@v0.3.0)
• JAX XLA (jax[cpu] == 0.3.24)
• PyTorch (torch == 1.13.0)
• DM-Tree (dm-tree == 0.1.7)

All results are reported on a workstation with an AMD Ryzen 9 5950X CPU @ 4.45GHz in an isolated virtual environment with Python 3.10.4. Run with the following command:

conda create --name optree-benchmark anaconda::python=3.10 --yes --no-default-packages
conda activate optree-benchmark
python3 -m pip install --editable '.[benchmark]' --extra-index-url https://download.pytorch.org/whl/cpu
python3 benchmark.py --number=10000 --repeat=5

The test inputs are nested containers (i.e., pytrees) extracted from torch.nn.Module objects. They are:

tiny_mlp = nn.Sequential(
    nn.Linear(1, 1, bias=True),
    nn.BatchNorm1d(1, affine=True, track_running_stats=True),
    nn.ReLU(),
    nn.Linear(1, 1, bias=False),
    nn.Sigmoid(),
)

and AlexNet, ResNet18, ResNet34, ResNet50, ResNet101, ResNet152, VisionTransformerH14 (ViT-H/14), and SwinTransformerB (Swin-B) from torchvsion. Please refer to benchmark.py for more details.

Tree Flatten

Module	Nodes	OpTree (μs)	JAX XLA (μs)	PyTorch (μs)	DM-Tree (μs)	Speedup (J / O)	Speedup (P / O)	Speedup (D / O)
TinyMLP	53	26.40	68.19	586.87	34.14	2.58	22.23	1.29
AlexNet	188	84.28	259.51	2182.07	125.12	3.08	25.89	1.48
ResNet18	698	288.57	807.27	7881.69	429.39	2.80	27.31	1.49
ResNet34	1242	580.75	1564.97	15082.84	819.02	2.69	25.97	1.41
ResNet50	1702	791.18	2081.17	20982.82	1104.62	2.63	26.52	1.40
ResNet101	3317	1603.93	3939.37	40382.14	2208.63	2.46	25.18	1.38
ResNet152	4932	2446.56	6267.98	56892.36	3139.17	2.56	23.25	1.28
ViT-H/14	3420	1681.48	4488.33	41703.16	2504.86	2.67	24.80	1.49
Swin-B	2881	1565.41	4091.10	34241.99	1936.75	2.61	21.87	1.24
					Average	2.68	24.78	1.38

Tree UnFlatten

Module	Nodes	OpTree (μs)	JAX XLA (μs)	PyTorch (μs)	DM-Tree (μs)	Speedup (J / O)	Speedup (P / O)	Speedup (D / O)
TinyMLP	53	59.47	163.00	257.56	967.00	2.74	4.33	16.26
AlexNet	188	234.68	701.56	1011.04	4000.43	2.99	4.31	17.05
ResNet18	698	758.82	2036.76	3391.87	12060.09	2.68	4.47	15.89
ResNet34	1242	1459.17	3886.79	6519.28	21435.14	2.66	4.47	14.69
ResNet50	1702	2003.60	5137.90	8341.17	29067.89	2.56	4.16	14.51
ResNet101	3317	4005.73	10203.31	17316.07	59531.47	2.55	4.32	14.86
ResNet152	4932	5644.08	15153.87	25438.67	88626.45	2.68	4.51	15.70
ViT-H/14	3420	4492.64	12544.41	18091.68	67876.19	2.79	4.03	15.11
Swin-B	2881	3637.86	9973.78	15353.31	57655.54	2.74	4.22	15.85
					Average	2.71	4.31	15.55

Tree Copy

Module	Nodes	OpTree (μs)	JAX XLA (μs)	PyTorch (μs)	DM-Tree (μs)	Speedup (J / O)	Speedup (P / O)	Speedup (D / O)
TinyMLP	53	90.43	234.08	871.84	1006.30	2.59	9.64	11.13
AlexNet	188	324.33	931.61	3263.23	4106.04	2.87	10.06	12.66
ResNet18	698	1111.82	2840.68	11836.55	12564.23	2.55	10.65	11.30
ResNet34	1242	2029.24	5129.39	20888.04	23559.50	2.53	10.29	11.61
ResNet50	1702	2884.39	7118.82	30239.69	29509.25	2.47	10.48	10.23
ResNet101	3317	5773.17	14396.40	60021.18	62725.03	2.49	10.40	10.86
ResNet152	4932	8552.95	21321.48	85857.53	86037.99	2.49	10.04	10.06
ViT-H/14	3420	6116.61	16038.69	59993.87	70215.65	2.62	9.81	11.48
Swin-B	2881	5466.03	14449.60	50528.12	60269.63	2.64	9.24	11.03
					Average	2.58	10.07	11.15

Tree Map

Module	Nodes	OpTree (μs)	JAX XLA (μs)	PyTorch (μs)	DM-Tree (μs)	Speedup (J / O)	Speedup (P / O)	Speedup (D / O)
TinyMLP	53	98.24	255.39	868.24	1032.07	2.60	8.84	10.51
AlexNet	188	337.87	1004.53	3304.64	4099.77	2.97	9.78	12.13
ResNet18	698	1171.69	3059.72	11921.16	12727.38	2.61	10.17	10.86
ResNet34	1242	2267.61	5793.53	22222.92	22437.44	2.55	9.80	9.89
ResNet50	1702	2961.05	7792.69	30132.32	31460.04	2.63	10.18	10.62
ResNet101	3317	6101.05	14342.22	56480.19	61830.65	2.35	9.26	10.13
ResNet152	4932	8568.48	21641.40	83021.19	87077.66	2.53	9.69	10.16
ViT-H/14	3420	6735.93	18027.05	63986.88	75742.33	2.68	9.50	11.24
Swin-B	2881	5756.71	14528.51	51052.90	60715.06	2.52	8.87	10.55
					Average	2.61	9.56	10.68

Tree Map (nargs)

Module	Nodes	OpTree (μs)	JAX XLA (μs)	PyTorch (μs)	DM-Tree (μs)	Speedup (J / O)	Speedup (P / O)	Speedup (D / O)
TinyMLP	53	144.61	391.05	N/A	3774.79	2.70	N/A	26.10
AlexNet	188	480.23	1515.41	N/A	15105.87	3.16	N/A	31.46
ResNet18	698	1690.19	4997.44	N/A	52089.06	2.96	N/A	30.82
ResNet34	1242	3084.36	8572.54	N/A	93923.27	2.78	N/A	30.45
ResNet50	1702	4441.17	11962.92	N/A	126937.65	2.69	N/A	28.58
ResNet101	3317	8155.78	22232.67	N/A	251333.88	2.73	N/A	30.82
ResNet152	4932	12862.88	33714.46	N/A	368424.63	2.62	N/A	28.64
ViT-H/14	3420	9511.10	27920.13	N/A	281245.95	2.94	N/A	29.57
Swin-B	2881	7628.29	22421.37	N/A	238211.56	2.94	N/A	31.23
					Average	2.83	N/A	29.74

TinyMLP(num_nodes=53, num_leaves=16, treespec=PyTreeSpec([OrderedDict([('tenso...), buffers=OrderedDict([])))])]))
| Subject          | OpTree (μs) | JAX XLA (μs) | PyTorch (μs) | DM-Tree (μs) | Speedup (J / O) | Speedup (P / O) | Speedup (D / O) |
| :--------------- | ----------: | -----------: | -----------: | -----------: | --------------: | --------------: | --------------: |
| Tree Flatten     |       26.40 |        68.19 |       586.87 |        34.14 |            2.58 |           22.23 |            1.29 |
| Tree UnFlatten   |       59.47 |       163.00 |       257.56 |       967.00 |            2.74 |            4.33 |           16.26 |
| Tree Copy        |       90.43 |       234.08 |       871.84 |      1006.30 |            2.59 |            9.64 |           11.13 |
| Tree Map         |       98.24 |       255.39 |       868.24 |      1032.07 |            2.60 |            8.84 |           10.51 |
| Tree Map (nargs) |      144.61 |       391.05 |          N/A |      3774.79 |            2.70 |             N/A |           26.10 |

### Check ###
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=False)[::-1]) == tree
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=True)[::-1]) == tree
✔ FLATTEN (OpTree vs. JAX XLA): optree.tree_leaves(tree, none_is_leaf=False) == jax.tree_util.tree_leaves(tree)
✔ FLATTEN (OpTree vs. PyTorch): optree.tree_leaves(tree, none_is_leaf=True) == torch_utils_pytree.tree_flatten(tree)[0]
✔ TREEMAP (OpTree vs. JAX XLA): optree.tree_map(fn, tree, none_is_leaf=False) == jax.tree_util.tree_map(fn, tree)
✔ TREEMAP (OpTree vs. PyTorch): optree.tree_map(fn, tree, none_is_leaf=True) == torch_utils_pytree.tree_map(fn, tree)

### Tree Flatten ###
~ OpTree :    26.40μs            <=  optree.tree_leaves(x)                      (None is Node)
✔ OpTree :    25.98μs -- x0.98   <=  optree.tree_leaves(x, none_is_leaf=False)  (None is Node)
~ OpTree :    26.19μs -- x0.99   <=  optree.tree_leaves(x, none_is_leaf=True)   (None is Leaf)
  JAX XLA:    68.19μs -- x2.58   <=  jax.tree_util.tree_leaves(x)
  PyTorch:   586.87μs -- x22.23  <=  torch_utils_pytree.tree_flatten(x)[0]
  DM-Tree:    34.14μs -- x1.29   <=  dm_tree.flatten(x)

### Tree UnFlatten ###
✔ OpTree :    59.47μs            <=  optree.tree_unflatten(spec, flat)  (None is Node)
~ OpTree :    59.71μs -- x1.00   <=  optree.tree_unflatten(spec, flat)  (None is Leaf)
  JAX XLA:   163.00μs -- x2.74   <=  jax.tree_util.tree_unflatten(spec, flat)
  PyTorch:   257.56μs -- x4.33   <=  torch_utils_pytree.tree_unflatten(flat, spec)
  DM-Tree:   967.00μs -- x16.26  <=  dm_tree.unflatten_as(spec, flat)

### Tree Copy ###
✔ OpTree :    90.43μs            <=  optree.tree_unflatten(*optree.tree_flatten(x)[::-1])                      (None is Node)
~ OpTree :    91.51μs -- x1.01   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=False)[::-1])  (None is Node)
~ OpTree :    91.34μs -- x1.01   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=True)[::-1])   (None is Leaf)
  JAX XLA:   234.08μs -- x2.59   <=  jax.tree_util.tree_unflatten(*jax.tree_util.tree_flatten(x)[::-1])
  PyTorch:   871.84μs -- x9.64   <=  torch_utils_pytree.tree_unflatten(*torch_utils_pytree.tree_flatten(x))
  DM-Tree:  1006.30μs -- x11.13  <=  dm_tree.unflatten_as(x, dm_tree.flatten(x))

### Tree Map ###
~ OpTree :    98.24μs            <=  optree.tree_map(fn1, x)                      (None is Node)
✔ OpTree :    97.62μs -- x0.99   <=  optree.tree_map(fn1, x, none_is_leaf=False)  (None is Node)
~ OpTree :    98.05μs -- x1.00   <=  optree.tree_map(fn1, x, none_is_leaf=True)   (None is Leaf)
  JAX XLA:   255.39μs -- x2.60   <=  jax.tree_util.tree_map(fn1, x)
  PyTorch:   868.24μs -- x8.84   <=  torch_utils_pytree.tree_map(fn1, x)
  DM-Tree:  1032.07μs -- x10.51  <=  dm_tree.map_structure(fn1, x)

### Tree Map (nargs) ###
~ OpTree :   144.61μs            <=  optree.tree_map(fn3, x, y, z)                      (None is Node)
✔ OpTree :   141.03μs -- x0.98   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=False)  (None is Node)
~ OpTree :   142.79μs -- x0.99   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=True)   (None is Leaf)
  JAX XLA:   391.05μs -- x2.70   <=  jax.tree_util.tree_map(fn3, x, y, z)
  DM-Tree:  3774.79μs -- x26.10  <=  dm_tree.map_structure_up_to(x, fn3, x, y, z)

AlexNet(num_nodes=188, num_leaves=32, treespec=PyTreeSpec(OrderedDict([('featur...]), buffers=OrderedDict([])))])))
| Subject          | OpTree (μs) | JAX XLA (μs) | PyTorch (μs) | DM-Tree (μs) | Speedup (J / O) | Speedup (P / O) | Speedup (D / O) |
| :--------------- | ----------: | -----------: | -----------: | -----------: | --------------: | --------------: | --------------: |
| Tree Flatten     |       84.28 |       259.51 |      2182.07 |       125.12 |            3.08 |           25.89 |            1.48 |
| Tree UnFlatten   |      234.68 |       701.56 |      1011.04 |      4000.43 |            2.99 |            4.31 |           17.05 |
| Tree Copy        |      324.33 |       931.61 |      3263.23 |      4106.04 |            2.87 |           10.06 |           12.66 |
| Tree Map         |      337.87 |      1004.53 |      3304.64 |      4099.77 |            2.97 |            9.78 |           12.13 |
| Tree Map (nargs) |      480.23 |      1515.41 |          N/A |     15105.87 |            3.16 |             N/A |           31.46 |

### Check ###
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=False)[::-1]) == tree
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=True)[::-1]) == tree
✔ FLATTEN (OpTree vs. JAX XLA): optree.tree_leaves(tree, none_is_leaf=False) == jax.tree_util.tree_leaves(tree)
✔ FLATTEN (OpTree vs. PyTorch): optree.tree_leaves(tree, none_is_leaf=True) == torch_utils_pytree.tree_flatten(tree)[0]
✔ TREEMAP (OpTree vs. JAX XLA): optree.tree_map(fn, tree, none_is_leaf=False) == jax.tree_util.tree_map(fn, tree)
✔ TREEMAP (OpTree vs. PyTorch): optree.tree_map(fn, tree, none_is_leaf=True) == torch_utils_pytree.tree_map(fn, tree)

### Tree Flatten ###
✔ OpTree :    84.28μs            <=  optree.tree_leaves(x)                      (None is Node)
~ OpTree :    84.64μs -- x1.00   <=  optree.tree_leaves(x, none_is_leaf=False)  (None is Node)
~ OpTree :    85.54μs -- x1.01   <=  optree.tree_leaves(x, none_is_leaf=True)   (None is Leaf)
  JAX XLA:   259.51μs -- x3.08   <=  jax.tree_util.tree_leaves(x)
  PyTorch:  2182.07μs -- x25.89  <=  torch_utils_pytree.tree_flatten(x)[0]
  DM-Tree:   125.12μs -- x1.48   <=  dm_tree.flatten(x)

### Tree UnFlatten ###
~ OpTree :   234.68μs            <=  optree.tree_unflatten(spec, flat)  (None is Node)
✔ OpTree :   234.02μs -- x1.00   <=  optree.tree_unflatten(spec, flat)  (None is Leaf)
  JAX XLA:   701.56μs -- x2.99   <=  jax.tree_util.tree_unflatten(spec, flat)
  PyTorch:  1011.04μs -- x4.31   <=  torch_utils_pytree.tree_unflatten(flat, spec)
  DM-Tree:  4000.43μs -- x17.05  <=  dm_tree.unflatten_as(spec, flat)

### Tree Copy ###
~ OpTree :   324.33μs            <=  optree.tree_unflatten(*optree.tree_flatten(x)[::-1])                      (None is Node)
~ OpTree :   324.09μs -- x1.00   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=False)[::-1])  (None is Node)
✔ OpTree :   323.18μs -- x1.00   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=True)[::-1])   (None is Leaf)
  JAX XLA:   931.61μs -- x2.87   <=  jax.tree_util.tree_unflatten(*jax.tree_util.tree_flatten(x)[::-1])
  PyTorch:  3263.23μs -- x10.06  <=  torch_utils_pytree.tree_unflatten(*torch_utils_pytree.tree_flatten(x))
  DM-Tree:  4106.04μs -- x12.66  <=  dm_tree.unflatten_as(x, dm_tree.flatten(x))

### Tree Map ###
✔ OpTree :   337.87μs            <=  optree.tree_map(fn1, x)                      (None is Node)
~ OpTree :   340.56μs -- x1.01   <=  optree.tree_map(fn1, x, none_is_leaf=False)  (None is Node)
~ OpTree :   338.36μs -- x1.00   <=  optree.tree_map(fn1, x, none_is_leaf=True)   (None is Leaf)
  JAX XLA:  1004.53μs -- x2.97   <=  jax.tree_util.tree_map(fn1, x)
  PyTorch:  3304.64μs -- x9.78   <=  torch_utils_pytree.tree_map(fn1, x)
  DM-Tree:  4099.77μs -- x12.13  <=  dm_tree.map_structure(fn1, x)

### Tree Map (nargs) ###
~ OpTree :   480.23μs            <=  optree.tree_map(fn3, x, y, z)                      (None is Node)
~ OpTree :   481.45μs -- x1.00   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=False)  (None is Node)
✔ OpTree :   479.35μs -- x1.00   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=True)   (None is Leaf)
  JAX XLA:  1515.41μs -- x3.16   <=  jax.tree_util.tree_map(fn3, x, y, z)
  DM-Tree: 15105.87μs -- x31.46  <=  dm_tree.map_structure_up_to(x, fn3, x, y, z)

ResNet18(num_nodes=698, num_leaves=244, treespec=PyTreeSpec(OrderedDict([('conv1'...]), buffers=OrderedDict([])))])))
| Subject          | OpTree (μs) | JAX XLA (μs) | PyTorch (μs) | DM-Tree (μs) | Speedup (J / O) | Speedup (P / O) | Speedup (D / O) |
| :--------------- | ----------: | -----------: | -----------: | -----------: | --------------: | --------------: | --------------: |
| Tree Flatten     |      288.57 |       807.27 |      7881.69 |       429.39 |            2.80 |           27.31 |            1.49 |
| Tree UnFlatten   |      758.82 |      2036.76 |      3391.87 |     12060.09 |            2.68 |            4.47 |           15.89 |
| Tree Copy        |     1111.82 |      2840.68 |     11836.55 |     12564.23 |            2.55 |           10.65 |           11.30 |
| Tree Map         |     1171.69 |      3059.72 |     11921.16 |     12727.38 |            2.61 |           10.17 |           10.86 |
| Tree Map (nargs) |     1690.19 |      4997.44 |          N/A |     52089.06 |            2.96 |             N/A |           30.82 |

### Check ###
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=False)[::-1]) == tree
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=True)[::-1]) == tree
✔ FLATTEN (OpTree vs. JAX XLA): optree.tree_leaves(tree, none_is_leaf=False) == jax.tree_util.tree_leaves(tree)
✔ FLATTEN (OpTree vs. PyTorch): optree.tree_leaves(tree, none_is_leaf=True) == torch_utils_pytree.tree_flatten(tree)[0]
✔ TREEMAP (OpTree vs. JAX XLA): optree.tree_map(fn, tree, none_is_leaf=False) == jax.tree_util.tree_map(fn, tree)
✔ TREEMAP (OpTree vs. PyTorch): optree.tree_map(fn, tree, none_is_leaf=True) == torch_utils_pytree.tree_map(fn, tree)

### Tree Flatten ###
✔ OpTree :   288.57μs            <=  optree.tree_leaves(x)                      (None is Node)
~ OpTree :   294.01μs -- x1.02   <=  optree.tree_leaves(x, none_is_leaf=False)  (None is Node)
~ OpTree :   294.99μs -- x1.02   <=  optree.tree_leaves(x, none_is_leaf=True)   (None is Leaf)
  JAX XLA:   807.27μs -- x2.80   <=  jax.tree_util.tree_leaves(x)
  PyTorch:  7881.69μs -- x27.31  <=  torch_utils_pytree.tree_flatten(x)[0]
  DM-Tree:   429.39μs -- x1.49   <=  dm_tree.flatten(x)

### Tree UnFlatten ###
✔ OpTree :   758.82μs            <=  optree.tree_unflatten(spec, flat)  (None is Node)
~ OpTree :   765.90μs -- x1.01   <=  optree.tree_unflatten(spec, flat)  (None is Leaf)
  JAX XLA:  2036.76μs -- x2.68   <=  jax.tree_util.tree_unflatten(spec, flat)
  PyTorch:  3391.87μs -- x4.47   <=  torch_utils_pytree.tree_unflatten(flat, spec)
  DM-Tree: 12060.09μs -- x15.89  <=  dm_tree.unflatten_as(spec, flat)

### Tree Copy ###
~ OpTree :  1111.82μs            <=  optree.tree_unflatten(*optree.tree_flatten(x)[::-1])                      (None is Node)
~ OpTree :  1103.80μs -- x0.99   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=False)[::-1])  (None is Node)
✔ OpTree :  1100.03μs -- x0.99   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=True)[::-1])   (None is Leaf)
  JAX XLA:  2840.68μs -- x2.55   <=  jax.tree_util.tree_unflatten(*jax.tree_util.tree_flatten(x)[::-1])
  PyTorch: 11836.55μs -- x10.65  <=  torch_utils_pytree.tree_unflatten(*torch_utils_pytree.tree_flatten(x))
  DM-Tree: 12564.23μs -- x11.30  <=  dm_tree.unflatten_as(x, dm_tree.flatten(x))

### Tree Map ###
✔ OpTree :  1171.69μs            <=  optree.tree_map(fn1, x)                      (None is Node)
~ OpTree :  1172.46μs -- x1.00   <=  optree.tree_map(fn1, x, none_is_leaf=False)  (None is Node)
~ OpTree :  1181.17μs -- x1.01   <=  optree.tree_map(fn1, x, none_is_leaf=True)   (None is Leaf)
  JAX XLA:  3059.72μs -- x2.61   <=  jax.tree_util.tree_map(fn1, x)
  PyTorch: 11921.16μs -- x10.17  <=  torch_utils_pytree.tree_map(fn1, x)
  DM-Tree: 12727.38μs -- x10.86  <=  dm_tree.map_structure(fn1, x)

### Tree Map (nargs) ###
✔ OpTree :  1690.19μs            <=  optree.tree_map(fn3, x, y, z)                      (None is Node)
~ OpTree :  1760.86μs -- x1.04   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=False)  (None is Node)
~ OpTree :  1761.76μs -- x1.04   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=True)   (None is Leaf)
  JAX XLA:  4997.44μs -- x2.96   <=  jax.tree_util.tree_map(fn3, x, y, z)
  DM-Tree: 52089.06μs -- x30.82  <=  dm_tree.map_structure_up_to(x, fn3, x, y, z)

ResNet34(num_nodes=1242, num_leaves=436, treespec=PyTreeSpec(OrderedDict([('conv1'...]), buffers=OrderedDict([])))])))
| Subject          | OpTree (μs) | JAX XLA (μs) | PyTorch (μs) | DM-Tree (μs) | Speedup (J / O) | Speedup (P / O) | Speedup (D / O) |
| :--------------- | ----------: | -----------: | -----------: | -----------: | --------------: | --------------: | --------------: |
| Tree Flatten     |      580.75 |      1564.97 |     15082.84 |       819.02 |            2.69 |           25.97 |            1.41 |
| Tree UnFlatten   |     1459.17 |      3886.79 |      6519.28 |     21435.14 |            2.66 |            4.47 |           14.69 |
| Tree Copy        |     2029.24 |      5129.39 |     20888.04 |     23559.50 |            2.53 |           10.29 |           11.61 |
| Tree Map         |     2267.61 |      5793.53 |     22222.92 |     22437.44 |            2.55 |            9.80 |            9.89 |
| Tree Map (nargs) |     3084.36 |      8572.54 |          N/A |     93923.27 |            2.78 |             N/A |           30.45 |

### Check ###
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=False)[::-1]) == tree
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=True)[::-1]) == tree
✔ FLATTEN (OpTree vs. JAX XLA): optree.tree_leaves(tree, none_is_leaf=False) == jax.tree_util.tree_leaves(tree)
✔ FLATTEN (OpTree vs. PyTorch): optree.tree_leaves(tree, none_is_leaf=True) == torch_utils_pytree.tree_flatten(tree)[0]
✔ TREEMAP (OpTree vs. JAX XLA): optree.tree_map(fn, tree, none_is_leaf=False) == jax.tree_util.tree_map(fn, tree)
✔ TREEMAP (OpTree vs. PyTorch): optree.tree_map(fn, tree, none_is_leaf=True) == torch_utils_pytree.tree_map(fn, tree)

### Tree Flatten ###
~ OpTree :   580.75μs            <=  optree.tree_leaves(x)                      (None is Node)
~ OpTree :   577.02μs -- x0.99   <=  optree.tree_leaves(x, none_is_leaf=False)  (None is Node)
✔ OpTree :   571.14μs -- x0.98   <=  optree.tree_leaves(x, none_is_leaf=True)   (None is Leaf)
  JAX XLA:  1564.97μs -- x2.69   <=  jax.tree_util.tree_leaves(x)
  PyTorch: 15082.84μs -- x25.97  <=  torch_utils_pytree.tree_flatten(x)[0]
  DM-Tree:   819.02μs -- x1.41   <=  dm_tree.flatten(x)

### Tree UnFlatten ###
✔ OpTree :  1459.17μs            <=  optree.tree_unflatten(spec, flat)  (None is Node)
~ OpTree :  1465.07μs -- x1.00   <=  optree.tree_unflatten(spec, flat)  (None is Leaf)
  JAX XLA:  3886.79μs -- x2.66   <=  jax.tree_util.tree_unflatten(spec, flat)
  PyTorch:  6519.28μs -- x4.47   <=  torch_utils_pytree.tree_unflatten(flat, spec)
  DM-Tree: 21435.14μs -- x14.69  <=  dm_tree.unflatten_as(spec, flat)

### Tree Copy ###
~ OpTree :  2029.24μs            <=  optree.tree_unflatten(*optree.tree_flatten(x)[::-1])                      (None is Node)
✔ OpTree :  2021.45μs -- x1.00   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=False)[::-1])  (None is Node)
~ OpTree :  2024.29μs -- x1.00   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=True)[::-1])   (None is Leaf)
  JAX XLA:  5129.39μs -- x2.53   <=  jax.tree_util.tree_unflatten(*jax.tree_util.tree_flatten(x)[::-1])
  PyTorch: 20888.04μs -- x10.29  <=  torch_utils_pytree.tree_unflatten(*torch_utils_pytree.tree_flatten(x))
  DM-Tree: 23559.50μs -- x11.61  <=  dm_tree.unflatten_as(x, dm_tree.flatten(x))

### Tree Map ###
~ OpTree :  2267.61μs            <=  optree.tree_map(fn1, x)                      (None is Node)
✔ OpTree :  2257.85μs -- x1.00   <=  optree.tree_map(fn1, x, none_is_leaf=False)  (None is Node)
~ OpTree :  2268.77μs -- x1.00   <=  optree.tree_map(fn1, x, none_is_leaf=True)   (None is Leaf)
  JAX XLA:  5793.53μs -- x2.55   <=  jax.tree_util.tree_map(fn1, x)
  PyTorch: 22222.92μs -- x9.80   <=  torch_utils_pytree.tree_map(fn1, x)
  DM-Tree: 22437.44μs -- x9.89   <=  dm_tree.map_structure(fn1, x)

### Tree Map (nargs) ###
~ OpTree :  3084.36μs            <=  optree.tree_map(fn3, x, y, z)                      (None is Node)
~ OpTree :  3080.29μs -- x1.00   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=False)  (None is Node)
✔ OpTree :  3072.45μs -- x1.00   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=True)   (None is Leaf)
  JAX XLA:  8572.54μs -- x2.78   <=  jax.tree_util.tree_map(fn3, x, y, z)
  DM-Tree: 93923.27μs -- x30.45  <=  dm_tree.map_structure_up_to(x, fn3, x, y, z)

ResNet50(num_nodes=1702, num_leaves=640, treespec=PyTreeSpec(OrderedDict([('conv1'...]), buffers=OrderedDict([])))])))
| Subject          | OpTree (μs) | JAX XLA (μs) | PyTorch (μs) | DM-Tree (μs) | Speedup (J / O) | Speedup (P / O) | Speedup (D / O) |
| :--------------- | ----------: | -----------: | -----------: | -----------: | --------------: | --------------: | --------------: |
| Tree Flatten     |      791.18 |      2081.17 |     20982.82 |      1104.62 |            2.63 |           26.52 |            1.40 |
| Tree UnFlatten   |     2003.60 |      5137.90 |      8341.17 |     29067.89 |            2.56 |            4.16 |           14.51 |
| Tree Copy        |     2884.39 |      7118.82 |     30239.69 |     29509.25 |            2.47 |           10.48 |           10.23 |
| Tree Map         |     2961.05 |      7792.69 |     30132.32 |     31460.04 |            2.63 |           10.18 |           10.62 |
| Tree Map (nargs) |     4441.17 |     11962.92 |          N/A |    126937.65 |            2.69 |             N/A |           28.58 |

### Check ###
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=False)[::-1]) == tree
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=True)[::-1]) == tree
✔ FLATTEN (OpTree vs. JAX XLA): optree.tree_leaves(tree, none_is_leaf=False) == jax.tree_util.tree_leaves(tree)
✔ FLATTEN (OpTree vs. PyTorch): optree.tree_leaves(tree, none_is_leaf=True) == torch_utils_pytree.tree_flatten(tree)[0]
✔ TREEMAP (OpTree vs. JAX XLA): optree.tree_map(fn, tree, none_is_leaf=False) == jax.tree_util.tree_map(fn, tree)
✔ TREEMAP (OpTree vs. PyTorch): optree.tree_map(fn, tree, none_is_leaf=True) == torch_utils_pytree.tree_map(fn, tree)

### Tree Flatten ###
~ OpTree :   791.18μs            <=  optree.tree_leaves(x)                      (None is Node)
~ OpTree :   791.38μs -- x1.00   <=  optree.tree_leaves(x, none_is_leaf=False)  (None is Node)
✔ OpTree :   779.75μs -- x0.99   <=  optree.tree_leaves(x, none_is_leaf=True)   (None is Leaf)
  JAX XLA:  2081.17μs -- x2.63   <=  jax.tree_util.tree_leaves(x)
  PyTorch: 20982.82μs -- x26.52  <=  torch_utils_pytree.tree_flatten(x)[0]
  DM-Tree:  1104.62μs -- x1.40   <=  dm_tree.flatten(x)

### Tree UnFlatten ###
~ OpTree :  2003.60μs            <=  optree.tree_unflatten(spec, flat)  (None is Node)
✔ OpTree :  2000.10μs -- x1.00   <=  optree.tree_unflatten(spec, flat)  (None is Leaf)
  JAX XLA:  5137.90μs -- x2.56   <=  jax.tree_util.tree_unflatten(spec, flat)
  PyTorch:  8341.17μs -- x4.16   <=  torch_utils_pytree.tree_unflatten(flat, spec)
  DM-Tree: 29067.89μs -- x14.51  <=  dm_tree.unflatten_as(spec, flat)

### Tree Copy ###
~ OpTree :  2884.39μs            <=  optree.tree_unflatten(*optree.tree_flatten(x)[::-1])                      (None is Node)
~ OpTree :  2879.97μs -- x1.00   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=False)[::-1])  (None is Node)
✔ OpTree :  2868.84μs -- x0.99   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=True)[::-1])   (None is Leaf)
  JAX XLA:  7118.82μs -- x2.47   <=  jax.tree_util.tree_unflatten(*jax.tree_util.tree_flatten(x)[::-1])
  PyTorch: 30239.69μs -- x10.48  <=  torch_utils_pytree.tree_unflatten(*torch_utils_pytree.tree_flatten(x))
  DM-Tree: 29509.25μs -- x10.23  <=  dm_tree.unflatten_as(x, dm_tree.flatten(x))

### Tree Map ###
✔ OpTree :  2961.05μs            <=  optree.tree_map(fn1, x)                      (None is Node)
~ OpTree :  3079.33μs -- x1.04   <=  optree.tree_map(fn1, x, none_is_leaf=False)  (None is Node)
~ OpTree :  3116.74μs -- x1.05   <=  optree.tree_map(fn1, x, none_is_leaf=True)   (None is Leaf)
  JAX XLA:  7792.69μs -- x2.63   <=  jax.tree_util.tree_map(fn1, x)
  PyTorch: 30132.32μs -- x10.18  <=  torch_utils_pytree.tree_map(fn1, x)
  DM-Tree: 31460.04μs -- x10.62  <=  dm_tree.map_structure(fn1, x)

### Tree Map (nargs) ###
~ OpTree :  4441.17μs            <=  optree.tree_map(fn3, x, y, z)                      (None is Node)
✔ OpTree :  4430.87μs -- x1.00   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=False)  (None is Node)
~ OpTree :  4449.43μs -- x1.00   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=True)   (None is Leaf)
  JAX XLA: 11962.92μs -- x2.69   <=  jax.tree_util.tree_map(fn3, x, y, z)
  DM-Tree: 126937.65μs -- x28.58  <=  dm_tree.map_structure_up_to(x, fn3, x, y, z)

ResNet101(num_nodes=3317, num_leaves=1252, treespec=PyTreeSpec(OrderedDict([('conv1'...]), buffers=OrderedDict([])))])))
| Subject          | OpTree (μs) | JAX XLA (μs) | PyTorch (μs) | DM-Tree (μs) | Speedup (J / O) | Speedup (P / O) | Speedup (D / O) |
| :--------------- | ----------: | -----------: | -----------: | -----------: | --------------: | --------------: | --------------: |
| Tree Flatten     |     1603.93 |      3939.37 |     40382.14 |      2208.63 |            2.46 |           25.18 |            1.38 |
| Tree UnFlatten   |     4005.73 |     10203.31 |     17316.07 |     59531.47 |            2.55 |            4.32 |           14.86 |
| Tree Copy        |     5773.17 |     14396.40 |     60021.18 |     62725.03 |            2.49 |           10.40 |           10.86 |
| Tree Map         |     6101.05 |     14342.22 |     56480.19 |     61830.65 |            2.35 |            9.26 |           10.13 |
| Tree Map (nargs) |     8155.78 |     22232.67 |          N/A |    251333.88 |            2.73 |             N/A |           30.82 |

### Check ###
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=False)[::-1]) == tree
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=True)[::-1]) == tree
✔ FLATTEN (OpTree vs. JAX XLA): optree.tree_leaves(tree, none_is_leaf=False) == jax.tree_util.tree_leaves(tree)
✔ FLATTEN (OpTree vs. PyTorch): optree.tree_leaves(tree, none_is_leaf=True) == torch_utils_pytree.tree_flatten(tree)[0]
✔ TREEMAP (OpTree vs. JAX XLA): optree.tree_map(fn, tree, none_is_leaf=False) == jax.tree_util.tree_map(fn, tree)
✔ TREEMAP (OpTree vs. PyTorch): optree.tree_map(fn, tree, none_is_leaf=True) == torch_utils_pytree.tree_map(fn, tree)

### Tree Flatten ###
~ OpTree :  1603.93μs            <=  optree.tree_leaves(x)                      (None is Node)
✔ OpTree :  1458.25μs -- x0.91   <=  optree.tree_leaves(x, none_is_leaf=False)  (None is Node)
~ OpTree :  1492.62μs -- x0.93   <=  optree.tree_leaves(x, none_is_leaf=True)   (None is Leaf)
  JAX XLA:  3939.37μs -- x2.46   <=  jax.tree_util.tree_leaves(x)
  PyTorch: 40382.14μs -- x25.18  <=  torch_utils_pytree.tree_flatten(x)[0]
  DM-Tree:  2208.63μs -- x1.38   <=  dm_tree.flatten(x)

### Tree UnFlatten ###
~ OpTree :  4005.73μs            <=  optree.tree_unflatten(spec, flat)  (None is Node)
✔ OpTree :  3957.47μs -- x0.99   <=  optree.tree_unflatten(spec, flat)  (None is Leaf)
  JAX XLA: 10203.31μs -- x2.55   <=  jax.tree_util.tree_unflatten(spec, flat)
  PyTorch: 17316.07μs -- x4.32   <=  torch_utils_pytree.tree_unflatten(flat, spec)
  DM-Tree: 59531.47μs -- x14.86  <=  dm_tree.unflatten_as(spec, flat)

### Tree Copy ###
~ OpTree :  5773.17μs            <=  optree.tree_unflatten(*optree.tree_flatten(x)[::-1])                      (None is Node)
✔ OpTree :  5741.73μs -- x0.99   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=False)[::-1])  (None is Node)
~ OpTree :  5759.01μs -- x1.00   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=True)[::-1])   (None is Leaf)
  JAX XLA: 14396.40μs -- x2.49   <=  jax.tree_util.tree_unflatten(*jax.tree_util.tree_flatten(x)[::-1])
  PyTorch: 60021.18μs -- x10.40  <=  torch_utils_pytree.tree_unflatten(*torch_utils_pytree.tree_flatten(x))
  DM-Tree: 62725.03μs -- x10.86  <=  dm_tree.unflatten_as(x, dm_tree.flatten(x))

### Tree Map ###
~ OpTree :  6101.05μs            <=  optree.tree_map(fn1, x)                      (None is Node)
~ OpTree :  6145.86μs -- x1.01   <=  optree.tree_map(fn1, x, none_is_leaf=False)  (None is Node)
✔ OpTree :  5709.67μs -- x0.94   <=  optree.tree_map(fn1, x, none_is_leaf=True)   (None is Leaf)
  JAX XLA: 14342.22μs -- x2.35   <=  jax.tree_util.tree_map(fn1, x)
  PyTorch: 56480.19μs -- x9.26   <=  torch_utils_pytree.tree_map(fn1, x)
  DM-Tree: 61830.65μs -- x10.13  <=  dm_tree.map_structure(fn1, x)

### Tree Map (nargs) ###
~ OpTree :  8155.78μs            <=  optree.tree_map(fn3, x, y, z)                      (None is Node)
~ OpTree :  8144.17μs -- x1.00   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=False)  (None is Node)
✔ OpTree :  8113.58μs -- x0.99   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=True)   (None is Leaf)
  JAX XLA: 22232.67μs -- x2.73   <=  jax.tree_util.tree_map(fn3, x, y, z)
  DM-Tree: 251333.88μs -- x30.82  <=  dm_tree.map_structure_up_to(x, fn3, x, y, z)

ResNet152(num_nodes=4932, num_leaves=1864, treespec=PyTreeSpec(OrderedDict([('conv1'...]), buffers=OrderedDict([])))])))
| Subject          | OpTree (μs) | JAX XLA (μs) | PyTorch (μs) | DM-Tree (μs) | Speedup (J / O) | Speedup (P / O) | Speedup (D / O) |
| :--------------- | ----------: | -----------: | -----------: | -----------: | --------------: | --------------: | --------------: |
| Tree Flatten     |     2446.56 |      6267.98 |     56892.36 |      3139.17 |            2.56 |           23.25 |            1.28 |
| Tree UnFlatten   |     5644.08 |     15153.87 |     25438.67 |     88626.45 |            2.68 |            4.51 |           15.70 |
| Tree Copy        |     8552.95 |     21321.48 |     85857.53 |     86037.99 |            2.49 |           10.04 |           10.06 |
| Tree Map         |     8568.48 |     21641.40 |     83021.19 |     87077.66 |            2.53 |            9.69 |           10.16 |
| Tree Map (nargs) |    12862.88 |     33714.46 |          N/A |    368424.63 |            2.62 |             N/A |           28.64 |

### Check ###
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=False)[::-1]) == tree
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=True)[::-1]) == tree
✔ FLATTEN (OpTree vs. JAX XLA): optree.tree_leaves(tree, none_is_leaf=False) == jax.tree_util.tree_leaves(tree)
✔ FLATTEN (OpTree vs. PyTorch): optree.tree_leaves(tree, none_is_leaf=True) == torch_utils_pytree.tree_flatten(tree)[0]
✔ TREEMAP (OpTree vs. JAX XLA): optree.tree_map(fn, tree, none_is_leaf=False) == jax.tree_util.tree_map(fn, tree)
✔ TREEMAP (OpTree vs. PyTorch): optree.tree_map(fn, tree, none_is_leaf=True) == torch_utils_pytree.tree_map(fn, tree)

### Tree Flatten ###
~ OpTree :  2446.56μs            <=  optree.tree_leaves(x)                      (None is Node)
~ OpTree :  2455.99μs -- x1.00   <=  optree.tree_leaves(x, none_is_leaf=False)  (None is Node)
✔ OpTree :  2429.96μs -- x0.99   <=  optree.tree_leaves(x, none_is_leaf=True)   (None is Leaf)
  JAX XLA:  6267.98μs -- x2.56   <=  jax.tree_util.tree_leaves(x)
  PyTorch: 56892.36μs -- x23.25  <=  torch_utils_pytree.tree_flatten(x)[0]
  DM-Tree:  3139.17μs -- x1.28   <=  dm_tree.flatten(x)

### Tree UnFlatten ###
✔ OpTree :  5644.08μs            <=  optree.tree_unflatten(spec, flat)  (None is Node)
~ OpTree :  5723.38μs -- x1.01   <=  optree.tree_unflatten(spec, flat)  (None is Leaf)
  JAX XLA: 15153.87μs -- x2.68   <=  jax.tree_util.tree_unflatten(spec, flat)
  PyTorch: 25438.67μs -- x4.51   <=  torch_utils_pytree.tree_unflatten(flat, spec)
  DM-Tree: 88626.45μs -- x15.70  <=  dm_tree.unflatten_as(spec, flat)

### Tree Copy ###
~ OpTree :  8552.95μs            <=  optree.tree_unflatten(*optree.tree_flatten(x)[::-1])                      (None is Node)
~ OpTree :  8531.50μs -- x1.00   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=False)[::-1])  (None is Node)
✔ OpTree :  8528.88μs -- x1.00   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=True)[::-1])   (None is Leaf)
  JAX XLA: 21321.48μs -- x2.49   <=  jax.tree_util.tree_unflatten(*jax.tree_util.tree_flatten(x)[::-1])
  PyTorch: 85857.53μs -- x10.04  <=  torch_utils_pytree.tree_unflatten(*torch_utils_pytree.tree_flatten(x))
  DM-Tree: 86037.99μs -- x10.06  <=  dm_tree.unflatten_as(x, dm_tree.flatten(x))

### Tree Map ###
~ OpTree :  8568.48μs            <=  optree.tree_map(fn1, x)                      (None is Node)
~ OpTree :  8569.48μs -- x1.00   <=  optree.tree_map(fn1, x, none_is_leaf=False)  (None is Node)
✔ OpTree :  8542.91μs -- x1.00   <=  optree.tree_map(fn1, x, none_is_leaf=True)   (None is Leaf)
  JAX XLA: 21641.40μs -- x2.53   <=  jax.tree_util.tree_map(fn1, x)
  PyTorch: 83021.19μs -- x9.69   <=  torch_utils_pytree.tree_map(fn1, x)
  DM-Tree: 87077.66μs -- x10.16  <=  dm_tree.map_structure(fn1, x)

### Tree Map (nargs) ###
~ OpTree : 12862.88μs            <=  optree.tree_map(fn3, x, y, z)                      (None is Node)
✔ OpTree : 12806.09μs -- x1.00   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=False)  (None is Node)
~ OpTree : 12909.94μs -- x1.00   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=True)   (None is Leaf)
  JAX XLA: 33714.46μs -- x2.62   <=  jax.tree_util.tree_map(fn3, x, y, z)
  DM-Tree: 368424.63μs -- x28.64  <=  dm_tree.map_structure_up_to(x, fn3, x, y, z)

ViT-H/14(num_nodes=3420, num_leaves=784, treespec=PyTreeSpec(OrderedDict([('conv_p...]), buffers=OrderedDict([])))])))
| Subject          | OpTree (μs) | JAX XLA (μs) | PyTorch (μs) | DM-Tree (μs) | Speedup (J / O) | Speedup (P / O) | Speedup (D / O) |
| :--------------- | ----------: | -----------: | -----------: | -----------: | --------------: | --------------: | --------------: |
| Tree Flatten     |     1681.48 |      4488.33 |     41703.16 |      2504.86 |            2.67 |           24.80 |            1.49 |
| Tree UnFlatten   |     4492.64 |     12544.41 |     18091.68 |     67876.19 |            2.79 |            4.03 |           15.11 |
| Tree Copy        |     6116.61 |     16038.69 |     59993.87 |     70215.65 |            2.62 |            9.81 |           11.48 |
| Tree Map         |     6735.93 |     18027.05 |     63986.88 |     75742.33 |            2.68 |            9.50 |           11.24 |
| Tree Map (nargs) |     9511.10 |     27920.13 |          N/A |    281245.95 |            2.94 |             N/A |           29.57 |

### Check ###
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=False)[::-1]) == tree
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=True)[::-1]) == tree
✔ FLATTEN (OpTree vs. JAX XLA): optree.tree_leaves(tree, none_is_leaf=False) == jax.tree_util.tree_leaves(tree)
✔ FLATTEN (OpTree vs. PyTorch): optree.tree_leaves(tree, none_is_leaf=True) == torch_utils_pytree.tree_flatten(tree)[0]
✔ TREEMAP (OpTree vs. JAX XLA): optree.tree_map(fn, tree, none_is_leaf=False) == jax.tree_util.tree_map(fn, tree)
✔ TREEMAP (OpTree vs. PyTorch): optree.tree_map(fn, tree, none_is_leaf=True) == torch_utils_pytree.tree_map(fn, tree)

### Tree Flatten ###
✔ OpTree :  1681.48μs            <=  optree.tree_leaves(x)                      (None is Node)
~ OpTree :  1702.21μs -- x1.01   <=  optree.tree_leaves(x, none_is_leaf=False)  (None is Node)
~ OpTree :  1694.58μs -- x1.01   <=  optree.tree_leaves(x, none_is_leaf=True)   (None is Leaf)
  JAX XLA:  4488.33μs -- x2.67   <=  jax.tree_util.tree_leaves(x)
  PyTorch: 41703.16μs -- x24.80  <=  torch_utils_pytree.tree_flatten(x)[0]
  DM-Tree:  2504.86μs -- x1.49   <=  dm_tree.flatten(x)

### Tree UnFlatten ###
✔ OpTree :  4492.64μs            <=  optree.tree_unflatten(spec, flat)  (None is Node)
~ OpTree :  4535.79μs -- x1.01   <=  optree.tree_unflatten(spec, flat)  (None is Leaf)
  JAX XLA: 12544.41μs -- x2.79   <=  jax.tree_util.tree_unflatten(spec, flat)
  PyTorch: 18091.68μs -- x4.03   <=  torch_utils_pytree.tree_unflatten(flat, spec)
  DM-Tree: 67876.19μs -- x15.11  <=  dm_tree.unflatten_as(spec, flat)

### Tree Copy ###
~ OpTree :  6116.61μs            <=  optree.tree_unflatten(*optree.tree_flatten(x)[::-1])                      (None is Node)
✔ OpTree :  6075.72μs -- x0.99   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=False)[::-1])  (None is Node)
~ OpTree :  6104.80μs -- x1.00   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=True)[::-1])   (None is Leaf)
  JAX XLA: 16038.69μs -- x2.62   <=  jax.tree_util.tree_unflatten(*jax.tree_util.tree_flatten(x)[::-1])
  PyTorch: 59993.87μs -- x9.81   <=  torch_utils_pytree.tree_unflatten(*torch_utils_pytree.tree_flatten(x))
  DM-Tree: 70215.65μs -- x11.48  <=  dm_tree.unflatten_as(x, dm_tree.flatten(x))

### Tree Map ###
~ OpTree :  6735.93μs            <=  optree.tree_map(fn1, x)                      (None is Node)
✔ OpTree :  6679.19μs -- x0.99   <=  optree.tree_map(fn1, x, none_is_leaf=False)  (None is Node)
~ OpTree :  6726.99μs -- x1.00   <=  optree.tree_map(fn1, x, none_is_leaf=True)   (None is Leaf)
  JAX XLA: 18027.05μs -- x2.68   <=  jax.tree_util.tree_map(fn1, x)
  PyTorch: 63986.88μs -- x9.50   <=  torch_utils_pytree.tree_map(fn1, x)
  DM-Tree: 75742.33μs -- x11.24  <=  dm_tree.map_structure(fn1, x)

### Tree Map (nargs) ###
~ OpTree :  9511.10μs            <=  optree.tree_map(fn3, x, y, z)                      (None is Node)
✔ OpTree :  9503.85μs -- x1.00   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=False)  (None is Node)
~ OpTree :  9550.25μs -- x1.00   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=True)   (None is Leaf)
  JAX XLA: 27920.13μs -- x2.94   <=  jax.tree_util.tree_map(fn3, x, y, z)
  DM-Tree: 281245.95μs -- x29.57  <=  dm_tree.map_structure_up_to(x, fn3, x, y, z)

Swin-B(num_nodes=2881, num_leaves=706, treespec=PyTreeSpec(OrderedDict([('featur...]), buffers=OrderedDict([])))])))
| Subject          | OpTree (μs) | JAX XLA (μs) | PyTorch (μs) | DM-Tree (μs) | Speedup (J / O) | Speedup (P / O) | Speedup (D / O) |
| :--------------- | ----------: | -----------: | -----------: | -----------: | --------------: | --------------: | --------------: |
| Tree Flatten     |     1565.41 |      4091.10 |     34241.99 |      1936.75 |            2.61 |           21.87 |            1.24 |
| Tree UnFlatten   |     3637.86 |      9973.78 |     15353.31 |     57655.54 |            2.74 |            4.22 |           15.85 |
| Tree Copy        |     5466.03 |     14449.60 |     50528.12 |     60269.63 |            2.64 |            9.24 |           11.03 |
| Tree Map         |     5756.71 |     14528.51 |     51052.90 |     60715.06 |            2.52 |            8.87 |           10.55 |
| Tree Map (nargs) |     7628.29 |     22421.37 |          N/A |    238211.56 |            2.94 |             N/A |           31.23 |

### Check ###
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=False)[::-1]) == tree
✔ COPY: optree.tree_unflatten(*optree.tree_flatten(tree, none_is_leaf=True)[::-1]) == tree
✔ FLATTEN (OpTree vs. JAX XLA): optree.tree_leaves(tree, none_is_leaf=False) == jax.tree_util.tree_leaves(tree)
✔ FLATTEN (OpTree vs. PyTorch): optree.tree_leaves(tree, none_is_leaf=True) == torch_utils_pytree.tree_flatten(tree)[0]
✔ TREEMAP (OpTree vs. JAX XLA): optree.tree_map(fn, tree, none_is_leaf=False) == jax.tree_util.tree_map(fn, tree)
✔ TREEMAP (OpTree vs. PyTorch): optree.tree_map(fn, tree, none_is_leaf=True) == torch_utils_pytree.tree_map(fn, tree)

### Tree Flatten ###
~ OpTree :  1565.41μs            <=  optree.tree_leaves(x)                      (None is Node)
~ OpTree :  1565.91μs -- x1.00   <=  optree.tree_leaves(x, none_is_leaf=False)  (None is Node)
✔ OpTree :  1550.64μs -- x0.99   <=  optree.tree_leaves(x, none_is_leaf=True)   (None is Leaf)
  JAX XLA:  4091.10μs -- x2.61   <=  jax.tree_util.tree_leaves(x)
  PyTorch: 34241.99μs -- x21.87  <=  torch_utils_pytree.tree_flatten(x)[0]
  DM-Tree:  1936.75μs -- x1.24   <=  dm_tree.flatten(x)

### Tree UnFlatten ###
~ OpTree :  3637.86μs            <=  optree.tree_unflatten(spec, flat)  (None is Node)
✔ OpTree :  3596.18μs -- x0.99   <=  optree.tree_unflatten(spec, flat)  (None is Leaf)
  JAX XLA:  9973.78μs -- x2.74   <=  jax.tree_util.tree_unflatten(spec, flat)
  PyTorch: 15353.31μs -- x4.22   <=  torch_utils_pytree.tree_unflatten(flat, spec)
  DM-Tree: 57655.54μs -- x15.85  <=  dm_tree.unflatten_as(spec, flat)

### Tree Copy ###
✔ OpTree :  5466.03μs            <=  optree.tree_unflatten(*optree.tree_flatten(x)[::-1])                      (None is Node)
~ OpTree :  5467.68μs -- x1.00   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=False)[::-1])  (None is Node)
~ OpTree :  5469.55μs -- x1.00   <=  optree.tree_unflatten(*optree.tree_flatten(x, none_is_leaf=True)[::-1])   (None is Leaf)
  JAX XLA: 14449.60μs -- x2.64   <=  jax.tree_util.tree_unflatten(*jax.tree_util.tree_flatten(x)[::-1])
  PyTorch: 50528.12μs -- x9.24   <=  torch_utils_pytree.tree_unflatten(*torch_utils_pytree.tree_flatten(x))
  DM-Tree: 60269.63μs -- x11.03  <=  dm_tree.unflatten_as(x, dm_tree.flatten(x))

### Tree Map ###
~ OpTree :  5756.71μs            <=  optree.tree_map(fn1, x)                      (None is Node)
~ OpTree :  5712.77μs -- x0.99   <=  optree.tree_map(fn1, x, none_is_leaf=False)  (None is Node)
✔ OpTree :  5706.22μs -- x0.99   <=  optree.tree_map(fn1, x, none_is_leaf=True)   (None is Leaf)
  JAX XLA: 14528.51μs -- x2.52   <=  jax.tree_util.tree_map(fn1, x)
  PyTorch: 51052.90μs -- x8.87   <=  torch_utils_pytree.tree_map(fn1, x)
  DM-Tree: 60715.06μs -- x10.55  <=  dm_tree.map_structure(fn1, x)

### Tree Map (nargs) ###
~ OpTree :  7628.29μs            <=  optree.tree_map(fn3, x, y, z)                      (None is Node)
~ OpTree :  7622.97μs -- x1.00   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=False)  (None is Node)
✔ OpTree :  7548.69μs -- x0.99   <=  optree.tree_map(fn3, x, y, z, none_is_leaf=True)   (None is Leaf)
  JAX XLA: 22421.37μs -- x2.94   <=  jax.tree_util.tree_map(fn3, x, y, z)
  DM-Tree: 238211.56μs -- x31.23  <=  dm_tree.map_structure_up_to(x, fn3, x, y, z)

---

License

OpTree is released under the Apache License 2.0.

OpTree is heavily based on JAX's implementation of the PyTree utility, with deep refactoring and several improvements. The original licenses can be found at JAX's Apache License 2.0 and Tensorflow's Apache License 2.0.