h4NSrSSKJr SSKJr SSKrSSKJr /SQr "SS\R5r \"S 5\R"S S 9SS j55r \R"SS S9SSj5r\"S 5\R"S S 9S55r\R"SS S9S5rg)uFunctions for encoding and decoding trees. Since a tree is a highly restricted form of graph, it can be represented concisely in several ways. This module includes functions for encoding and decoding trees in the form of nested tuples and Prüfer sequences. The former requires a rooted tree, whereas the latter can be applied to unrooted trees. Furthermore, there is a bijection from Prüfer sequences to labeled trees. )Counter)chainN)not_implemented_for)from_nested_tuplefrom_prufer_sequenceNotATreeto_nested_tupleto_prufer_sequencec\rSrSrSrSrg)rzRaised when a function expects a tree (that is, a connected undirected graph with no cycles) but gets a non-tree graph as input instead. N)__name__ __module__ __qualname____firstlineno____doc____static_attributes__r Q/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/tree/coding.pyrrsrrdirectedT)graphsc^^UU4Sjm[R"U5(d[R"S5eX;a[R"SUSU35eT"XS5$)aReturns a nested tuple representation of the given tree. The nested tuple representation of a tree is defined recursively. The tree with one node and no edges is represented by the empty tuple, ``()``. A tree with ``k`` subtrees is represented by a tuple of length ``k`` in which each element is the nested tuple representation of a subtree. Parameters ---------- T : NetworkX graph An undirected graph object representing a tree. root : node The node in ``T`` to interpret as the root of the tree. canonical_form : bool If ``True``, each tuple is sorted so that the function returns a canonical form for rooted trees. This means "lighter" subtrees will appear as nested tuples before "heavier" subtrees. In this way, each isomorphic rooted tree has the same nested tuple representation. Returns ------- tuple A nested tuple representation of the tree. Notes ----- This function is *not* the inverse of :func:`from_nested_tuple`; the only guarantee is that the rooted trees are isomorphic. See also -------- from_nested_tuple to_prufer_sequence Examples -------- The tree need not be a balanced binary tree:: >>> T = nx.Graph() >>> T.add_edges_from([(0, 1), (0, 2), (0, 3)]) >>> T.add_edges_from([(1, 4), (1, 5)]) >>> T.add_edges_from([(3, 6), (3, 7)]) >>> root = 0 >>> nx.to_nested_tuple(T, root) (((), ()), (), ((), ())) Continuing the above example, if ``canonical_form`` is ``True``, the nested tuples will be sorted:: >>> nx.to_nested_tuple(T, root, canonical_form=True) ((), ((), ()), ((), ())) Even the path graph can be interpreted as a tree:: >>> T = nx.path_graph(4) >>> root = 0 >>> nx.to_nested_tuple(T, root) ((((),),),) c>^^[TT5U1- n[U5S:XagUUU4SjU5nT(a [U5n[U5$)aDRecursively compute the nested tuple representation of the given rooted tree. ``_parent`` is the parent node of ``root`` in the supertree in which ``T`` is a subtree, or ``None`` if ``root`` is the root of the supertree. This argument is used to determine which neighbors of ``root`` are children and which is the parent. rr c38># UHnT"TUT5v M g7fNr ).0vr _make_tupleroots r 7to_nested_tuple.._make_tuple..vs<8a+aD))8s)setlensortedtuple)rr _parentchildrennestedrcanonical_forms`` rr$to_nested_tuple.._make_tuplegsHqw<7)+ x=A <8< F^FV}rprovided graph is not a treezGraph z contains no node N)nxis_treer NodeNotFound)rr r*rs `@rr r #sWH* ::a==kk899 }ooqc);D6BCC q %%rT)r returns_graphc^U4SjmT"U5nU(a^[S/S[R"US555n[U5VVs0sHupEXT_M nnn[R"X&5nU$s snnf)aReturns the rooted tree corresponding to the given nested tuple. The nested tuple representation of a tree is defined recursively. The tree with one node and no edges is represented by the empty tuple, ``()``. A tree with ``k`` subtrees is represented by a tuple of length ``k`` in which each element is the nested tuple representation of a subtree. Parameters ---------- sequence : tuple A nested tuple representing a rooted tree. sensible_relabeling : bool Whether to relabel the nodes of the tree so that nodes are labeled in increasing order according to their breadth-first search order from the root node. Returns ------- NetworkX graph The tree corresponding to the given nested tuple, whose root node is node 0. If ``sensible_labeling`` is ``True``, nodes will be labeled in breadth-first search order starting from the root node. Notes ----- This function is *not* the inverse of :func:`to_nested_tuple`; the only guarantee is that the rooted trees are isomorphic. See also -------- to_nested_tuple from_prufer_sequence Examples -------- Sensible relabeling ensures that the nodes are labeled from the root starting at 0:: >>> balanced = (((), ()), ((), ())) >>> T = nx.from_nested_tuple(balanced, sensible_relabeling=True) >>> edges = [(0, 1), (0, 2), (1, 3), (1, 4), (2, 5), (2, 6)] >>> all((u, v) in T.edges() or (v, u) in T.edges() for (u, v) in edges) True c>[U5S:Xa[R"S5$[RR UVs/sH nT"U5S4PM sn5$s snf)zRecursively creates a tree from the given sequence of nested tuples. This function employs the :func:`~networkx.tree.join` function to recursively join subtrees into a larger tree. r)r$r- empty_graphtree join_trees)sequencechild _make_trees rr9%from_nested_tuple.._make_treesS x=A >>!$ $ww!!x"PxeJu$5q#9x"PQQ"PsArc3*# UH upUv M g7frr )rurs rr!$from_nested_tuple..sA.@da.@s)rr- bfs_edges enumerate relabel_nodes)r7sensible_relabelingr bfs_nodesirlabelsr9s @rrrsufR* 8A1#Abll1a.@AB #,Y#78#741!$#78   Q ' H 9sA7cx^^ [T5nUS:aSn[R"U5e[R"T5(d[R"S5e[ T5[ [ U55:wa [S5e[TR55m UU 4Sjn[U 4Sj[ U555=pE/n[ US- 5H^nU"U5nURU5 T U==S-ss'X:a T US:XaUnM;[U 4Sj[ US-U555=pEM` U$) u;Returns the Prüfer sequence of the given tree. A *Prüfer sequence* is a list of *n* - 2 numbers between 0 and *n* - 1, inclusive. The tree corresponding to a given Prüfer sequence can be recovered by repeatedly joining a node in the sequence with a node with the smallest potential degree according to the sequence. Parameters ---------- T : NetworkX graph An undirected graph object representing a tree. Returns ------- list The Prüfer sequence of the given tree. Raises ------ NetworkXPointlessConcept If the number of nodes in `T` is less than two. NotATree If `T` is not a tree. KeyError If the set of nodes in `T` is not {0, …, *n* - 1}. Notes ----- There is a bijection from labeled trees to Prüfer sequences. This function is the inverse of the :func:`from_prufer_sequence` function. Sometimes Prüfer sequences use nodes labeled from 1 to *n* instead of from 0 to *n* - 1. This function requires nodes to be labeled in the latter form. You can use :func:`~networkx.relabel_nodes` to relabel the nodes of your tree to the appropriate format. This implementation is from [1]_ and has a running time of $O(n)$. See also -------- to_nested_tuple from_prufer_sequence References ---------- .. [1] Wang, Xiaodong, Lei Wang, and Yingjie Wu. "An optimal algorithm for Prufer codes." *Journal of Software Engineering and Applications* 2.02 (2009): 111. Examples -------- There is a bijection between Prüfer sequences and labeled trees, so this function is the inverse of the :func:`from_prufer_sequence` function: >>> edges = [(0, 3), (1, 3), (2, 3), (3, 4), (4, 5)] >>> tree = nx.Graph(edges) >>> sequence = nx.to_prufer_sequence(tree) >>> sequence [3, 3, 3, 4] >>> tree2 = nx.from_prufer_sequence(sequence) >>> list(tree2.edges()) == edges True u>Prüfer sequence undefined for trees with fewer than two nodesr,z*tree must have node labels {0, ..., n - 1}c4>[U4SjTU55$)Nc3>># UHnTUS:dMUv M g7fr3Nr )rrdegrees rr!6to_prufer_sequence..parents../s5t!vay1}AAt  )next)r<rrJs rparents#to_prufer_sequence..parents.s5qt555rc3>># UHnTUS:XdMUv M g7frIr rkrJs rr!%to_prufer_sequence..1;1F1INQQrLr3c3>># UHnTUS:XdMUv M g7frIr rQs rr!rS:N(;1vayA~QQ(;rL) r$r-NetworkXPointlessConceptr.rr#rangeKeyErrordictrJrMappend) rnmsgrNindexr<resultrCrrJs ` @rr r s V AA1uN))#.. ::a==kk899 1vU1XCDD !((* F6;a;;;E F 1q5\ AJ aq Q 9aANeai(;NN NEA MrcX^[U5S-n[[U[U555m[R "U5n[ 5n[U4Sj[U555=pEUHnUS:dXaS- :a[R"SUS- SU35eURXV5 URU5 TU==S-ss'Xd:a TUS:XaUnMq[U4Sj[US-U555=pEM [ U5U- nUupVURXV5 U$)uReturns the tree corresponding to the given Prüfer sequence. A *Prüfer sequence* is a list of *n* - 2 numbers between 0 and *n* - 1, inclusive. The tree corresponding to a given Prüfer sequence can be recovered by repeatedly joining a node in the sequence with a node with the smallest potential degree according to the sequence. Parameters ---------- sequence : list A Prüfer sequence, which is a list of *n* - 2 integers between zero and *n* - 1, inclusive. Returns ------- NetworkX graph The tree corresponding to the given Prüfer sequence. Raises ------ NetworkXError If the Prüfer sequence is not valid. Notes ----- There is a bijection from labeled trees to Prüfer sequences. This function is the inverse of the :func:`from_prufer_sequence` function. Sometimes Prüfer sequences use nodes labeled from 1 to *n* instead of from 0 to *n* - 1. This function requires nodes to be labeled in the latter form. You can use :func:`networkx.relabel_nodes` to relabel the nodes of your tree to the appropriate format. This implementation is from [1]_ and has a running time of $O(n)$. References ---------- .. [1] Wang, Xiaodong, Lei Wang, and Yingjie Wu. "An optimal algorithm for Prufer codes." *Journal of Software Engineering and Applications* 2.02 (2009): 111. See also -------- from_nested_tuple to_prufer_sequence Examples -------- There is a bijection between Prüfer sequences and labeled trees, so this function is the inverse of the :func:`to_prufer_sequence` function: >>> edges = [(0, 3), (1, 3), (2, 3), (3, 4), (4, 5)] >>> tree = nx.Graph(edges) >>> sequence = nx.to_prufer_sequence(tree) >>> sequence [3, 3, 3, 4] >>> tree2 = nx.from_prufer_sequence(sequence) >>> list(tree2.edges()) == edges True rFc3>># UHnTUS:XdMUv M g7frIr rQs rr!'from_prufer_sequence..rTrLrr3z6Invalid Prufer sequence: Values must be between 0 and z, got c3>># UHnTUS:XdMUv M g7frIr rQs rr!rbrVrL) r$rrrXr-r4r#rM NetworkXErroradd_edgeadd) r7r\r not_orphanedr^r<rorphansrJs @rrr>s F H AU8U1X. /F qA5L;a;;;E  q5AAI""H1VTUSVW  1q Q 9aANeai(;NN NEA!f|#G DAJJq Hr)F)r collectionsr itertoolsrnetworkxr-networkx.utilsr__all__NetworkXExceptionr _dispatchabler rr rr rrrps  . r##Z \&!\&~T2P 3P fZ a!aHT2^ 3^ r