# Source code for networkx.generators.trees

```
"""Functions for generating trees."""
from collections import defaultdict
import networkx as nx
from networkx.utils import py_random_state
__all__ = ["prefix_tree", "random_tree", "prefix_tree_recursive"]
[docs]def prefix_tree(paths):
"""Creates a directed prefix tree from a list of paths.
Usually the paths are described as strings or lists of integers.
A "prefix tree" represents the prefix structure of the strings.
Each node represents a prefix of some string. The root represents
the empty prefix with children for the single letter prefixes which
in turn have children for each double letter prefix starting with
the single letter corresponding to the parent node, and so on.
More generally the prefixes do not need to be strings. A prefix refers
to the start of a sequence. The root has children for each one element
prefix and they have children for each two element prefix that starts
with the one element sequence of the parent, and so on.
Note that this implementation uses integer nodes with an attribute.
Each node has an attribute "source" whose value is the original element
of the path to which this node corresponds. For example, suppose `paths`
consists of one path: "can". Then the nodes `[1, 2, 3]` which represent
this path have "source" values "c", "a" and "n".
All the descendants of a node have a common prefix in the sequence/path
associated with that node. From the returned tree, the prefix for each
node can be constructed by traversing the tree up to the root and
accumulating the "source" values along the way.
The root node is always `0` and has "source" attribute `None`.
The root is the only node with in-degree zero.
The nil node is always `-1` and has "source" attribute `"NIL"`.
The nil node is the only node with out-degree zero.
Parameters
----------
paths: iterable of paths
An iterable of paths which are themselves sequences.
Matching prefixes among these sequences are identified with
nodes of the prefix tree. One leaf of the tree is associated
with each path. (Identical paths are associated with the same
leaf of the tree.)
Returns
-------
tree: DiGraph
A directed graph representing an arborescence consisting of the
prefix tree generated by `paths`. Nodes are directed "downward",
from parent to child. A special "synthetic" root node is added
to be the parent of the first node in each path. A special
"synthetic" leaf node, the "nil" node `-1`, is added to be the child
of all nodes representing the last element in a path. (The
addition of this nil node technically makes this not an
arborescence but a directed acyclic graph; removing the nil node
makes it an arborescence.)
Notes
-----
The prefix tree is also known as a *trie*.
Examples
--------
Create a prefix tree from a list of strings with common prefixes::
>>> paths = ["ab", "abs", "ad"]
>>> T = nx.prefix_tree(paths)
>>> list(T.edges)
[(0, 1), (1, 2), (1, 4), (2, -1), (2, 3), (3, -1), (4, -1)]
The leaf nodes can be obtained as predecessors of the nil node::
>>> root, NIL = 0, -1
>>> list(T.predecessors(NIL))
[2, 3, 4]
To recover the original paths that generated the prefix tree,
traverse up the tree from the node `-1` to the node `0`::
>>> recovered = []
>>> for v in T.predecessors(NIL):
... prefix = ""
... while v != root:
... prefix = str(T.nodes[v]["source"]) + prefix
... v = next(T.predecessors(v)) # only one predecessor
... recovered.append(prefix)
>>> sorted(recovered)
['ab', 'abs', 'ad']
"""
def get_children(parent, paths):
children = defaultdict(list)
# Populate dictionary with key(s) as the child/children of the root and
# value(s) as the remaining paths of the corresponding child/children
for path in paths:
# If path is empty, we add an edge to the NIL node.
if not path:
tree.add_edge(parent, NIL)
continue
child, *rest = path
# `child` may exist as the head of more than one path in `paths`.
children[child].append(rest)
return children
# Initialize the prefix tree with a root node and a nil node.
tree = nx.DiGraph()
root = 0
tree.add_node(root, source=None)
NIL = -1
tree.add_node(NIL, source="NIL")
children = get_children(root, paths)
stack = [(root, iter(children.items()))]
while stack:
parent, remaining_children = stack[-1]
try:
child, remaining_paths = next(remaining_children)
# Pop item off stack if there are no remaining children
except StopIteration:
stack.pop()
continue
# We relabel each child with an unused name.
new_name = len(tree) - 1
# The "source" node attribute stores the original node name.
tree.add_node(new_name, source=child)
tree.add_edge(parent, new_name)
children = get_children(new_name, remaining_paths)
stack.append((new_name, iter(children.items())))
return tree
def prefix_tree_recursive(paths):
"""Recursively creates a directed prefix tree from a list of paths.
The original recursive version of prefix_tree for comparison. It is
the same algorithm but the recursion is unrolled onto a stack.
Usually the paths are described as strings or lists of integers.
A "prefix tree" represents the prefix structure of the strings.
Each node represents a prefix of some string. The root represents
the empty prefix with children for the single letter prefixes which
in turn have children for each double letter prefix starting with
the single letter corresponding to the parent node, and so on.
More generally the prefixes do not need to be strings. A prefix refers
to the start of a sequence. The root has children for each one element
prefix and they have children for each two element prefix that starts
with the one element sequence of the parent, and so on.
Note that this implementation uses integer nodes with an attribute.
Each node has an attribute "source" whose value is the original element
of the path to which this node corresponds. For example, suppose `paths`
consists of one path: "can". Then the nodes `[1, 2, 3]` which represent
this path have "source" values "c", "a" and "n".
All the descendants of a node have a common prefix in the sequence/path
associated with that node. From the returned tree, ehe prefix for each
node can be constructed by traversing the tree up to the root and
accumulating the "source" values along the way.
The root node is always `0` and has "source" attribute `None`.
The root is the only node with in-degree zero.
The nil node is always `-1` and has "source" attribute `"NIL"`.
The nil node is the only node with out-degree zero.
Parameters
----------
paths: iterable of paths
An iterable of paths which are themselves sequences.
Matching prefixes among these sequences are identified with
nodes of the prefix tree. One leaf of the tree is associated
with each path. (Identical paths are associated with the same
leaf of the tree.)
Returns
-------
tree: DiGraph
A directed graph representing an arborescence consisting of the
prefix tree generated by `paths`. Nodes are directed "downward",
from parent to child. A special "synthetic" root node is added
to be the parent of the first node in each path. A special
"synthetic" leaf node, the "nil" node `-1`, is added to be the child
of all nodes representing the last element in a path. (The
addition of this nil node technically makes this not an
arborescence but a directed acyclic graph; removing the nil node
makes it an arborescence.)
Notes
-----
The prefix tree is also known as a *trie*.
Examples
--------
Create a prefix tree from a list of strings with common prefixes::
>>> paths = ["ab", "abs", "ad"]
>>> T = nx.prefix_tree(paths)
>>> list(T.edges)
[(0, 1), (1, 2), (1, 4), (2, -1), (2, 3), (3, -1), (4, -1)]
The leaf nodes can be obtained as predecessors of the nil node.
>>> root, NIL = 0, -1
>>> list(T.predecessors(NIL))
[2, 3, 4]
To recover the original paths that generated the prefix tree,
traverse up the tree from the node `-1` to the node `0`::
>>> recovered = []
>>> for v in T.predecessors(NIL):
... prefix = ""
... while v != root:
... prefix = str(T.nodes[v]["source"]) + prefix
... v = next(T.predecessors(v)) # only one predecessor
... recovered.append(prefix)
>>> sorted(recovered)
['ab', 'abs', 'ad']
"""
def _helper(paths, root, tree):
"""Recursively create a trie from the given list of paths.
`paths` is a list of paths, each of which is itself a list of
nodes, relative to the given `root` (but not including it). This
list of paths will be interpreted as a tree-like structure, in
which two paths that share a prefix represent two branches of
the tree with the same initial segment.
`root` is the parent of the node at index 0 in each path.
`tree` is the "accumulator", the :class:`networkx.DiGraph`
representing the branching to which the new nodes and edges will
be added.
"""
# For each path, remove the first node and make it a child of root.
# Any remaining paths then get processed recursively.
children = defaultdict(list)
for path in paths:
# If path is empty, we add an edge to the NIL node.
if not path:
tree.add_edge(root, NIL)
continue
child, *rest = path
# `child` may exist as the head of more than one path in `paths`.
children[child].append(rest)
# Add a node for each child, connect root, recurse to remaining paths
for child, remaining_paths in children.items():
# We relabel each child with an unused name.
new_name = len(tree) - 1
# The "source" node attribute stores the original node name.
tree.add_node(new_name, source=child)
tree.add_edge(root, new_name)
_helper(remaining_paths, new_name, tree)
# Initialize the prefix tree with a root node and a nil node.
tree = nx.DiGraph()
root = 0
tree.add_node(root, source=None)
NIL = -1
tree.add_node(NIL, source="NIL")
# Populate the tree.
_helper(paths, root, tree)
return tree
# From the Wikipedia article on Prüfer sequences:
#
# > Generating uniformly distributed random Prüfer sequences and
# > converting them into the corresponding trees is a straightforward
# > method of generating uniformly distributed random labelled trees.
#
[docs]@py_random_state(1)
def random_tree(n, seed=None, create_using=None):
"""Returns a uniformly random tree on `n` nodes.
Parameters
----------
n : int
A positive integer representing the number of nodes in the tree.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
NetworkX graph
A tree, given as an undirected graph, whose nodes are numbers in
the set {0, …, *n* - 1}.
Raises
------
NetworkXPointlessConcept
If `n` is zero (because the null graph is not a tree).
Notes
-----
The current implementation of this function generates a uniformly
random Prüfer sequence then converts that to a tree via the
:func:`~networkx.from_prufer_sequence` function. Since there is a
bijection between Prüfer sequences of length *n* - 2 and trees on
*n* nodes, the tree is chosen uniformly at random from the set of
all trees on *n* nodes.
Examples
--------
>>> tree = nx.random_tree(n=10, seed=0)
>>> print(nx.forest_str(tree, sources=[0]))
╙── 0
├── 3
└── 4
├── 6
│ ├── 1
│ ├── 2
│ └── 7
│ └── 8
│ └── 5
└── 9
>>> tree = nx.random_tree(n=10, seed=0, create_using=nx.DiGraph)
>>> print(nx.forest_str(tree))
╙── 0
├─╼ 3
└─╼ 4
├─╼ 6
│ ├─╼ 1
│ ├─╼ 2
│ └─╼ 7
│ └─╼ 8
│ └─╼ 5
└─╼ 9
"""
if n == 0:
raise nx.NetworkXPointlessConcept("the null graph is not a tree")
# Cannot create a Prüfer sequence unless `n` is at least two.
if n == 1:
utree = nx.empty_graph(1, create_using)
else:
sequence = [seed.choice(range(n)) for i in range(n - 2)]
utree = nx.from_prufer_sequence(sequence)
if create_using is None:
tree = utree
else:
tree = nx.empty_graph(0, create_using)
if tree.is_directed():
# Use a arbitrary root node and dfs to define edge directions
edges = nx.dfs_edges(utree, source=0)
else:
edges = utree.edges
# Populate the specified graph type
tree.add_nodes_from(utree.nodes)
tree.add_edges_from(edges)
return tree
```