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<h1>Source code for networkx.algorithms.tree.coding</h1><div class="highlight"><pre>
<span></span><span class="sd">"""Functions for encoding and decoding trees.</span>
<span class="sd">Since a tree is a highly restricted form of graph, it can be represented</span>
<span class="sd">concisely in several ways. This module includes functions for encoding</span>
<span class="sd">and decoding trees in the form of nested tuples and Prüfer</span>
<span class="sd">sequences. The former requires a rooted tree, whereas the latter can be</span>
<span class="sd">applied to unrooted trees. Furthermore, there is a bijection from Prüfer</span>
<span class="sd">sequences to labeled trees.</span>
<span class="sd">"""</span>
<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">Counter</span>
<span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">chain</span>
<span class="kn">import</span> <span class="nn">networkx</span> <span class="k">as</span> <span class="nn">nx</span>
<span class="kn">from</span> <span class="nn">networkx.utils</span> <span class="kn">import</span> <span class="n">not_implemented_for</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
<span class="s2">"from_nested_tuple"</span><span class="p">,</span>
<span class="s2">"from_prufer_sequence"</span><span class="p">,</span>
<span class="s2">"NotATree"</span><span class="p">,</span>
<span class="s2">"to_nested_tuple"</span><span class="p">,</span>
<span class="s2">"to_prufer_sequence"</span><span class="p">,</span>
<span class="p">]</span>
<div class="viewcode-block" id="NotATree"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.tree.coding.NotATree.html#networkx.algorithms.tree.coding.NotATree">[docs]</a><span class="k">class</span> <span class="nc">NotATree</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">NetworkXException</span><span class="p">):</span>
<span class="sd">"""Raised when a function expects a tree (that is, a connected</span>
<span class="sd"> undirected graph with no cycles) but gets a non-tree graph as input</span>
<span class="sd"> instead.</span>
<span class="sd"> """</span></div>
<div class="viewcode-block" id="to_nested_tuple"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.tree.coding.to_nested_tuple.html#networkx.algorithms.tree.coding.to_nested_tuple">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"directed"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">to_nested_tuple</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">root</span><span class="p">,</span> <span class="n">canonical_form</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
<span class="sd">"""Returns a nested tuple representation of the given tree.</span>
<span class="sd"> The nested tuple representation of a tree is defined</span>
<span class="sd"> recursively. The tree with one node and no edges is represented by</span>
<span class="sd"> the empty tuple, ``()``. A tree with ``k`` subtrees is represented</span>
<span class="sd"> by a tuple of length ``k`` in which each element is the nested tuple</span>
<span class="sd"> representation of a subtree.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> T : NetworkX graph</span>
<span class="sd"> An undirected graph object representing a tree.</span>
<span class="sd"> root : node</span>
<span class="sd"> The node in ``T`` to interpret as the root of the tree.</span>
<span class="sd"> canonical_form : bool</span>
<span class="sd"> If ``True``, each tuple is sorted so that the function returns</span>
<span class="sd"> a canonical form for rooted trees. This means "lighter" subtrees</span>
<span class="sd"> will appear as nested tuples before "heavier" subtrees. In this</span>
<span class="sd"> way, each isomorphic rooted tree has the same nested tuple</span>
<span class="sd"> representation.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> tuple</span>
<span class="sd"> A nested tuple representation of the tree.</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This function is *not* the inverse of :func:`from_nested_tuple`; the</span>
<span class="sd"> only guarantee is that the rooted trees are isomorphic.</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> from_nested_tuple</span>
<span class="sd"> to_prufer_sequence</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> The tree need not be a balanced binary tree::</span>
<span class="sd"> >>> T = nx.Graph()</span>
<span class="sd"> >>> T.add_edges_from([(0, 1), (0, 2), (0, 3)])</span>
<span class="sd"> >>> T.add_edges_from([(1, 4), (1, 5)])</span>
<span class="sd"> >>> T.add_edges_from([(3, 6), (3, 7)])</span>
<span class="sd"> >>> root = 0</span>
<span class="sd"> >>> nx.to_nested_tuple(T, root)</span>
<span class="sd"> (((), ()), (), ((), ()))</span>
<span class="sd"> Continuing the above example, if ``canonical_form`` is ``True``, the</span>
<span class="sd"> nested tuples will be sorted::</span>
<span class="sd"> >>> nx.to_nested_tuple(T, root, canonical_form=True)</span>
<span class="sd"> ((), ((), ()), ((), ()))</span>
<span class="sd"> Even the path graph can be interpreted as a tree::</span>
<span class="sd"> >>> T = nx.path_graph(4)</span>
<span class="sd"> >>> root = 0</span>
<span class="sd"> >>> nx.to_nested_tuple(T, root)</span>
<span class="sd"> ((((),),),)</span>
<span class="sd"> """</span>
<span class="k">def</span> <span class="nf">_make_tuple</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">root</span><span class="p">,</span> <span class="n">_parent</span><span class="p">):</span>
<span class="sd">"""Recursively compute the nested tuple representation of the</span>
<span class="sd"> given rooted tree.</span>
<span class="sd"> ``_parent`` is the parent node of ``root`` in the supertree in</span>
<span class="sd"> which ``T`` is a subtree, or ``None`` if ``root`` is the root of</span>
<span class="sd"> the supertree. This argument is used to determine which</span>
<span class="sd"> neighbors of ``root`` are children and which is the parent.</span>
<span class="sd"> """</span>
<span class="c1"># Get the neighbors of `root` that are not the parent node. We</span>
<span class="c1"># are guaranteed that `root` is always in `T` by construction.</span>
<span class="n">children</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">T</span><span class="p">[</span><span class="n">root</span><span class="p">])</span> <span class="o">-</span> <span class="p">{</span><span class="n">_parent</span><span class="p">}</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">children</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="p">()</span>
<span class="n">nested</span> <span class="o">=</span> <span class="p">(</span><span class="n">_make_tuple</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">root</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">children</span><span class="p">)</span>
<span class="k">if</span> <span class="n">canonical_form</span><span class="p">:</span>
<span class="n">nested</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">nested</span><span class="p">)</span>
<span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">nested</span><span class="p">)</span>
<span class="c1"># Do some sanity checks on the input.</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_tree</span><span class="p">(</span><span class="n">T</span><span class="p">):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NotATree</span><span class="p">(</span><span class="s2">"provided graph is not a tree"</span><span class="p">)</span>
<span class="k">if</span> <span class="n">root</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">T</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NodeNotFound</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Graph </span><span class="si">{</span><span class="n">T</span><span class="si">}</span><span class="s2"> contains no node </span><span class="si">{</span><span class="n">root</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
<span class="k">return</span> <span class="n">_make_tuple</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">root</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span></div>
<div class="viewcode-block" id="from_nested_tuple"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.tree.coding.from_nested_tuple.html#networkx.algorithms.tree.coding.from_nested_tuple">[docs]</a><span class="k">def</span> <span class="nf">from_nested_tuple</span><span class="p">(</span><span class="n">sequence</span><span class="p">,</span> <span class="n">sensible_relabeling</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
<span class="sd">"""Returns the rooted tree corresponding to the given nested tuple.</span>
<span class="sd"> The nested tuple representation of a tree is defined</span>
<span class="sd"> recursively. The tree with one node and no edges is represented by</span>
<span class="sd"> the empty tuple, ``()``. A tree with ``k`` subtrees is represented</span>
<span class="sd"> by a tuple of length ``k`` in which each element is the nested tuple</span>
<span class="sd"> representation of a subtree.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> sequence : tuple</span>
<span class="sd"> A nested tuple representing a rooted tree.</span>
<span class="sd"> sensible_relabeling : bool</span>
<span class="sd"> Whether to relabel the nodes of the tree so that nodes are</span>
<span class="sd"> labeled in increasing order according to their breadth-first</span>
<span class="sd"> search order from the root node.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> NetworkX graph</span>
<span class="sd"> The tree corresponding to the given nested tuple, whose root</span>
<span class="sd"> node is node 0. If ``sensible_labeling`` is ``True``, nodes will</span>
<span class="sd"> be labeled in breadth-first search order starting from the root</span>
<span class="sd"> node.</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This function is *not* the inverse of :func:`to_nested_tuple`; the</span>
<span class="sd"> only guarantee is that the rooted trees are isomorphic.</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> to_nested_tuple</span>
<span class="sd"> from_prufer_sequence</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> Sensible relabeling ensures that the nodes are labeled from the root</span>
<span class="sd"> starting at 0::</span>
<span class="sd"> >>> balanced = (((), ()), ((), ()))</span>
<span class="sd"> >>> T = nx.from_nested_tuple(balanced, sensible_relabeling=True)</span>
<span class="sd"> >>> edges = [(0, 1), (0, 2), (1, 3), (1, 4), (2, 5), (2, 6)]</span>
<span class="sd"> >>> all((u, v) in T.edges() or (v, u) in T.edges() for (u, v) in edges)</span>
<span class="sd"> True</span>
<span class="sd"> """</span>
<span class="k">def</span> <span class="nf">_make_tree</span><span class="p">(</span><span class="n">sequence</span><span class="p">):</span>
<span class="sd">"""Recursively creates a tree from the given sequence of nested</span>
<span class="sd"> tuples.</span>
<span class="sd"> This function employs the :func:`~networkx.tree.join` function</span>
<span class="sd"> to recursively join subtrees into a larger tree.</span>
<span class="sd"> """</span>
<span class="c1"># The empty sequence represents the empty tree, which is the</span>
<span class="c1"># (unique) graph with a single node. We mark the single node</span>
<span class="c1"># with an attribute that indicates that it is the root of the</span>
<span class="c1"># graph.</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">sequence</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="n">nx</span><span class="o">.</span><span class="n">empty_graph</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="c1"># For a nonempty sequence, get the subtrees for each child</span>
<span class="c1"># sequence and join all the subtrees at their roots. After</span>
<span class="c1"># joining the subtrees, the root is node 0.</span>
<span class="k">return</span> <span class="n">nx</span><span class="o">.</span><span class="n">tree</span><span class="o">.</span><span class="n">join</span><span class="p">([(</span><span class="n">_make_tree</span><span class="p">(</span><span class="n">child</span><span class="p">),</span> <span class="mi">0</span><span class="p">)</span> <span class="k">for</span> <span class="n">child</span> <span class="ow">in</span> <span class="n">sequence</span><span class="p">])</span>
<span class="c1"># Make the tree and remove the `is_root` node attribute added by the</span>
<span class="c1"># helper function.</span>
<span class="n">T</span> <span class="o">=</span> <span class="n">_make_tree</span><span class="p">(</span><span class="n">sequence</span><span class="p">)</span>
<span class="k">if</span> <span class="n">sensible_relabeling</span><span class="p">:</span>
<span class="c1"># Relabel the nodes according to their breadth-first search</span>
<span class="c1"># order, starting from the root node (that is, the node 0).</span>
<span class="n">bfs_nodes</span> <span class="o">=</span> <span class="n">chain</span><span class="p">([</span><span class="mi">0</span><span class="p">],</span> <span class="p">(</span><span class="n">v</span> <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">nx</span><span class="o">.</span><span class="n">bfs_edges</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="mi">0</span><span class="p">)))</span>
<span class="n">labels</span> <span class="o">=</span> <span class="p">{</span><span class="n">v</span><span class="p">:</span> <span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">bfs_nodes</span><span class="p">)}</span>
<span class="c1"># We would like to use `copy=False`, but `relabel_nodes` doesn't</span>
<span class="c1"># allow a relabel mapping that can't be topologically sorted.</span>
<span class="n">T</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">relabel_nodes</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">labels</span><span class="p">)</span>
<span class="k">return</span> <span class="n">T</span></div>
<div class="viewcode-block" id="to_prufer_sequence"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.tree.coding.to_prufer_sequence.html#networkx.algorithms.tree.coding.to_prufer_sequence">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"directed"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">to_prufer_sequence</span><span class="p">(</span><span class="n">T</span><span class="p">):</span>
<span class="sa">r</span><span class="sd">"""Returns the Prüfer sequence of the given tree.</span>
<span class="sd"> A *Prüfer sequence* is a list of *n* - 2 numbers between 0 and</span>
<span class="sd"> *n* - 1, inclusive. The tree corresponding to a given Prüfer</span>
<span class="sd"> sequence can be recovered by repeatedly joining a node in the</span>
<span class="sd"> sequence with a node with the smallest potential degree according to</span>
<span class="sd"> the sequence.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> T : NetworkX graph</span>
<span class="sd"> An undirected graph object representing a tree.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> list</span>
<span class="sd"> The Prüfer sequence of the given tree.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXPointlessConcept</span>
<span class="sd"> If the number of nodes in `T` is less than two.</span>
<span class="sd"> NotATree</span>
<span class="sd"> If `T` is not a tree.</span>
<span class="sd"> KeyError</span>
<span class="sd"> If the set of nodes in `T` is not {0, …, *n* - 1}.</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> There is a bijection from labeled trees to Prüfer sequences. This</span>
<span class="sd"> function is the inverse of the :func:`from_prufer_sequence`</span>
<span class="sd"> function.</span>
<span class="sd"> Sometimes Prüfer sequences use nodes labeled from 1 to *n* instead</span>
<span class="sd"> of from 0 to *n* - 1. This function requires nodes to be labeled in</span>
<span class="sd"> the latter form. You can use :func:`~networkx.relabel_nodes` to</span>
<span class="sd"> relabel the nodes of your tree to the appropriate format.</span>
<span class="sd"> This implementation is from [1]_ and has a running time of</span>
<span class="sd"> $O(n)$.</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> to_nested_tuple</span>
<span class="sd"> from_prufer_sequence</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Wang, Xiaodong, Lei Wang, and Yingjie Wu.</span>
<span class="sd"> "An optimal algorithm for Prufer codes."</span>
<span class="sd"> *Journal of Software Engineering and Applications* 2.02 (2009): 111.</span>
<span class="sd"> <https://doi.org/10.4236/jsea.2009.22016></span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> There is a bijection between Prüfer sequences and labeled trees, so</span>
<span class="sd"> this function is the inverse of the :func:`from_prufer_sequence`</span>
<span class="sd"> function:</span>
<span class="sd"> >>> edges = [(0, 3), (1, 3), (2, 3), (3, 4), (4, 5)]</span>
<span class="sd"> >>> tree = nx.Graph(edges)</span>
<span class="sd"> >>> sequence = nx.to_prufer_sequence(tree)</span>
<span class="sd"> >>> sequence</span>
<span class="sd"> [3, 3, 3, 4]</span>
<span class="sd"> >>> tree2 = nx.from_prufer_sequence(sequence)</span>
<span class="sd"> >>> list(tree2.edges()) == edges</span>
<span class="sd"> True</span>
<span class="sd"> """</span>
<span class="c1"># Perform some sanity checks on the input.</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">T</span><span class="p">)</span>
<span class="k">if</span> <span class="n">n</span> <span class="o"><</span> <span class="mi">2</span><span class="p">:</span>
<span class="n">msg</span> <span class="o">=</span> <span class="s2">"Prüfer sequence undefined for trees with fewer than two nodes"</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXPointlessConcept</span><span class="p">(</span><span class="n">msg</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_tree</span><span class="p">(</span><span class="n">T</span><span class="p">):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NotATree</span><span class="p">(</span><span class="s2">"provided graph is not a tree"</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">set</span><span class="p">(</span><span class="n">T</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">set</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)):</span>
<span class="k">raise</span> <span class="ne">KeyError</span><span class="p">(</span><span class="s2">"tree must have node labels {0, ..., n - 1}"</span><span class="p">)</span>
<span class="n">degree</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">T</span><span class="o">.</span><span class="n">degree</span><span class="p">())</span>
<span class="k">def</span> <span class="nf">parents</span><span class="p">(</span><span class="n">u</span><span class="p">):</span>
<span class="k">return</span> <span class="nb">next</span><span class="p">(</span><span class="n">v</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">T</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="k">if</span> <span class="n">degree</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">></span> <span class="mi">1</span><span class="p">)</span>
<span class="n">index</span> <span class="o">=</span> <span class="n">u</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">k</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">if</span> <span class="n">degree</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="p">):</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">parents</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
<span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="n">degree</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
<span class="k">if</span> <span class="n">v</span> <span class="o"><</span> <span class="n">index</span> <span class="ow">and</span> <span class="n">degree</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">v</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">index</span> <span class="o">=</span> <span class="n">u</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">k</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">index</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="k">if</span> <span class="n">degree</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">return</span> <span class="n">result</span></div>
<div class="viewcode-block" id="from_prufer_sequence"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.tree.coding.from_prufer_sequence.html#networkx.algorithms.tree.coding.from_prufer_sequence">[docs]</a><span class="k">def</span> <span class="nf">from_prufer_sequence</span><span class="p">(</span><span class="n">sequence</span><span class="p">):</span>
<span class="sa">r</span><span class="sd">"""Returns the tree corresponding to the given Prüfer sequence.</span>
<span class="sd"> A *Prüfer sequence* is a list of *n* - 2 numbers between 0 and</span>
<span class="sd"> *n* - 1, inclusive. The tree corresponding to a given Prüfer</span>
<span class="sd"> sequence can be recovered by repeatedly joining a node in the</span>
<span class="sd"> sequence with a node with the smallest potential degree according to</span>
<span class="sd"> the sequence.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> sequence : list</span>
<span class="sd"> A Prüfer sequence, which is a list of *n* - 2 integers between</span>
<span class="sd"> zero and *n* - 1, inclusive.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> NetworkX graph</span>
<span class="sd"> The tree corresponding to the given Prüfer sequence.</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> There is a bijection from labeled trees to Prüfer sequences. This</span>
<span class="sd"> function is the inverse of the :func:`from_prufer_sequence` function.</span>
<span class="sd"> Sometimes Prüfer sequences use nodes labeled from 1 to *n* instead</span>
<span class="sd"> of from 0 to *n* - 1. This function requires nodes to be labeled in</span>
<span class="sd"> the latter form. You can use :func:`networkx.relabel_nodes` to</span>
<span class="sd"> relabel the nodes of your tree to the appropriate format.</span>
<span class="sd"> This implementation is from [1]_ and has a running time of</span>
<span class="sd"> $O(n)$.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Wang, Xiaodong, Lei Wang, and Yingjie Wu.</span>
<span class="sd"> "An optimal algorithm for Prufer codes."</span>
<span class="sd"> *Journal of Software Engineering and Applications* 2.02 (2009): 111.</span>
<span class="sd"> <https://doi.org/10.4236/jsea.2009.22016></span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> from_nested_tuple</span>
<span class="sd"> to_prufer_sequence</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> There is a bijection between Prüfer sequences and labeled trees, so</span>
<span class="sd"> this function is the inverse of the :func:`to_prufer_sequence`</span>
<span class="sd"> function:</span>
<span class="sd"> >>> edges = [(0, 3), (1, 3), (2, 3), (3, 4), (4, 5)]</span>
<span class="sd"> >>> tree = nx.Graph(edges)</span>
<span class="sd"> >>> sequence = nx.to_prufer_sequence(tree)</span>
<span class="sd"> >>> sequence</span>
<span class="sd"> [3, 3, 3, 4]</span>
<span class="sd"> >>> tree2 = nx.from_prufer_sequence(sequence)</span>
<span class="sd"> >>> list(tree2.edges()) == edges</span>
<span class="sd"> True</span>
<span class="sd"> """</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">sequence</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span>
<span class="c1"># `degree` stores the remaining degree (plus one) for each node. The</span>
<span class="c1"># degree of a node in the decoded tree is one more than the number</span>
<span class="c1"># of times it appears in the code.</span>
<span class="n">degree</span> <span class="o">=</span> <span class="n">Counter</span><span class="p">(</span><span class="n">chain</span><span class="p">(</span><span class="n">sequence</span><span class="p">,</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span>
<span class="n">T</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">empty_graph</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="c1"># `not_orphaned` is the set of nodes that have a parent in the</span>
<span class="c1"># tree. After the loop, there should be exactly two nodes that are</span>
<span class="c1"># not in this set.</span>
<span class="n">not_orphaned</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
<span class="n">index</span> <span class="o">=</span> <span class="n">u</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">k</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">if</span> <span class="n">degree</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">sequence</span><span class="p">:</span>
<span class="n">T</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="n">not_orphaned</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
<span class="n">degree</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
<span class="k">if</span> <span class="n">v</span> <span class="o"><</span> <span class="n">index</span> <span class="ow">and</span> <span class="n">degree</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">v</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">index</span> <span class="o">=</span> <span class="n">u</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">k</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">index</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="k">if</span> <span class="n">degree</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
<span class="c1"># At this point, there must be exactly two orphaned nodes; join them.</span>
<span class="n">orphans</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">T</span><span class="p">)</span> <span class="o">-</span> <span class="n">not_orphaned</span>
<span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">orphans</span>
<span class="n">T</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="k">return</span> <span class="n">T</span></div>
</pre></div>
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