text: #-*- coding: utf-8 -*-
  """
  Recognition Tests
  =================
  
  A *forest* is an acyclic, undirected graph, and a *tree* is a connected forest.
  Depending on the subfield, there are various conventions for generalizing these
  definitions to directed graphs.
  
  In one convention, directed variants of forest and tree are defined in an
  identical manner, except that the direction of the edges is ignored. In effect,
  each directed edge is treated as a single undirected edge. Then, additional
  restrictions are imposed to define *branchings* and *arborescences*.
  
  In another convention, directed variants of forest and tree correspond to
  the previous convention's branchings and arborescences, respectively. Then two
  new terms, *polyforest* and *polytree*, are defined to correspond to the other
  convention's forest and tree.
  
  Summarizing::
  
     +-----------------------------+
     | Convention A | Convention B |
     +=============================+
     | forest       | polyforest   |
     | tree         | polytree     |
     | branching    | forest       |
     | arborescence | tree         |
     +-----------------------------+
  
  Each convention has its reasons. The first convention emphasizes definitional
  similarity in that directed forests and trees are only concerned with
  acyclicity and do not have an in-degree constraint, just as their undirected
  counterparts do not. The second convention emphasizes functional similarity
  in the sense that the directed analog of a spanning tree is a spanning
  arborescence. That is, take any spanning tree and choose one node as the root.
  Then every edge is assigned a direction such there is a directed path from the
  root to every other node. The result is a spanning arborescence.
  
  NetworkX follows convention "A". Explicitly, these are:
  
  undirected forest
     An undirected graph with no undirected cycles.
  
  undirected tree
     A connected, undirected forest.
  
  directed forest
     A directed graph with no undirected cycles. Equivalently, the underlying
     graph structure (which ignores edge orientations) is an undirected forest.
     In convention B, this is known as a polyforest.
  
  directed tree
     A weakly connected, directed forest. Equivalently, the underlying graph
     structure (which ignores edge orientations) is an undirected tree. In
     convention B, this is known as a polytree.
  
  branching
     A directed forest with each node having, at most, one parent. So the maximum
     in-degree is equal to 1. In convention B, this is known as a forest.
  
  arborescence
     A directed tree with each node having, at most, one parent. So the maximum
     in-degree is equal to 1. In convention B, this is known as a tree.
  
  For trees and arborescences, the adjective "spanning" may be added to designate
  that the graph, when considered as a forest/branching, consists of a single
  tree/arborescence that includes all nodes in the graph. It is true, by
  definition, that every tree/arborescence is spanning with respect to the nodes
  that define the tree/arborescence and so, it might seem redundant to introduce
  the notion of "spanning". However, the nodes may represent a subset of
  nodes from a larger graph, and it is in this context that the term "spanning"
  becomes a useful notion.
  
  """
  
  import networkx as nx
  
  __author__ = """\n""".join([
      'Ferdinando Papale <ferdinando.papale@gmail.com>',
      'chebee7i <chebee7i@gmail.com>',
  ])
  
  
  __all__ = ['is_arborescence', 'is_branching', 'is_forest', 'is_tree']
  
  
  @nx.utils.not_implemented_for('undirected')
  def is_arborescence(G):
      """
      Returns True if `G` is an arborescence.
  
      An arborescence is a directed tree with maximum in-degree equal to 1.
  
      Parameters
      ----------
      G : graph
          The graph to test.
  
      Returns
      -------
      b : bool
          A boolean that is True if `G` is an arborescence.
  
      Notes
      -----
      In another convention, an arborescence is known as a *tree*.
  
      See Also
      --------
      is_tree
  
      """
      return is_tree(G) and max(d for n, d in G.in_degree()) <= 1
  
  
  @nx.utils.not_implemented_for('undirected')
  def is_branching(G):
      """
      Returns True if `G` is a branching.
  
      A branching is a directed forest with maximum in-degree equal to 1.
  
      Parameters
      ----------
      G : directed graph
          The directed graph to test.
  
      Returns
      -------
      b : bool
          A boolean that is True if `G` is a branching.
  
      Notes
      -----
      In another convention, a branching is also known as a *forest*.
  
      See Also
      --------
      is_forest
  
      """
      return is_forest(G) and max(d for n, d in G.in_degree()) <= 1
  
  
  def is_forest(G):
      """
      Returns True if `G` is a forest.
  
      A forest is a graph with no undirected cycles.
  
      For directed graphs, `G` is a forest if the underlying graph is a forest.
      The underlying graph is obtained by treating each directed edge as a single
      undirected edge in a multigraph.
  
      Parameters
      ----------
      G : graph
          The graph to test.
  
      Returns
      -------
      b : bool
          A boolean that is True if `G` is a forest.
  
      Notes
      -----
      In another convention, a directed forest is known as a *polyforest* and
      then *forest* corresponds to a *branching*.
  
      See Also
      --------
      is_branching
  
      """
      if len(G) == 0:
          raise nx.exception.NetworkXPointlessConcept('G has no nodes.')
  
      if G.is_directed():
          components = (G.subgraph(c) for c in nx.weakly_connected_components(G))
      else:
          components = (G.subgraph(c) for c in nx.connected_components(G))
  
      return all(len(c) - 1 == c.number_of_edges() for c in components)
  
  
  def is_tree(G):
      """
      Returns True if `G` is a tree.
  
      A tree is a connected graph with no undirected cycles.
  
      For directed graphs, `G` is a tree if the underlying graph is a tree. The
      underlying graph is obtained by treating each directed edge as a single
      undirected edge in a multigraph.
  
      Parameters
      ----------
      G : graph
          The graph to test.
  
      Returns
      -------
      b : bool
          A boolean that is True if `G` is a tree.
  
      Notes
      -----
      In another convention, a directed tree is known as a *polytree* and then
      *tree* corresponds to an *arborescence*.
  
      See Also
      --------
      is_arborescence
  
      """
      if len(G) == 0:
          raise nx.exception.NetworkXPointlessConcept('G has no nodes.')
  
      if G.is_directed():
          is_connected = nx.is_weakly_connected
      else:
          is_connected = nx.is_connected
  
      # A connected graph with no cycles has n-1 edges.
      return len(G) - 1 == G.number_of_edges() and is_connected(G)
  