*h/rSSKJr SSKJr SSKJr SSKJr SSKJ r J r SSK J r SSK Jr "SS \5rg ) )Basic)Tuple)Array_sympify)flatteniterable)as_int) defaultdictc\rSrSrSrSrSrSrSr\ S5r \ S5r \ S5r \ S5r \ S5r\S 5r\S 5r\S 5rS r\S 5rSrSSjrSSjrSrg)Prufer ap The Prufer correspondence is an algorithm that describes the bijection between labeled trees and the Prufer code. A Prufer code of a labeled tree is unique up to isomorphism and has a length of n - 2. Prufer sequences were first used by Heinz Prufer to give a proof of Cayley's formula. References ========== .. [1] https://mathworld.wolfram.com/LabeledTree.html NcURc.URURSSUR5UlUR$)aReturns Prufer sequence for the Prufer object. This sequence is found by removing the highest numbered vertex, recording the node it was attached to, and continuing until only two vertices remain. The Prufer sequence is the list of recorded nodes. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).prufer_repr [3, 3, 3, 4] >>> Prufer([1, 0, 0]).prufer_repr [1, 0, 0] See Also ======== to_prufer N) _prufer_repr to_prufer _tree_reprnodesselfs W/home/mande/repo/quber/.venv/lib/python3.13/site-packages/sympy/combinatorics/prufer.py prufer_reprPrufer.prufer_repr s<.    $ $tq/A4:: ND    czURc#URURSS5UlUR$)aBReturns the tree representation of the Prufer object. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).tree_repr [[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]] >>> Prufer([1, 0, 0]).tree_repr [[1, 2], [0, 1], [0, 3], [0, 4]] See Also ======== to_tree N)rto_treerrs r tree_reprPrufer.tree_repr;s3& ?? ""ll4+<+>> from sympy.combinatorics.prufer import Prufer >>> Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).nodes 6 >>> Prufer([1, 0, 0]).nodes 5 )_nodesrs rr Prufer.nodesRs{{rc^URcUR5UlUR$)aReturns the rank of the Prufer sequence. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> p = Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]) >>> p.rank 778 >>> p.next(1).rank 779 >>> p.prev().rank 777 See Also ======== prufer_rank, next, prev, size )_rank prufer_rankrs rrank Prufer.rankbs(, :: ))+DJzzrcnURUR5R5RS-$)zReturn the number of possible trees of this Prufer object. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> Prufer([0]*4).size == Prufer([6]*4).size == 1296 True See Also ======== prufer_rank, rank, next, prev )prevr$rs rsize Prufer.size|s+"yy#((*//!33rc[[5n/nUH!nX$S==S- ss'X$S==S- ss'M# [US- 5Hn[U5H nX&S:XdM O SnUH$nWUS:XaUSnO XdS:XaUSnUcM$ O URU5 WU4H+nX(==S-ss'X((aMUR U5 M- UR W5 M U$)a}Return the Prufer sequence for a tree given as a list of edges where ``n`` is the number of nodes in the tree. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> a = Prufer([[0, 1], [0, 2], [0, 3]]) >>> a.prufer_repr [0, 0] >>> Prufer.to_prufer([[0, 1], [0, 2], [0, 3]], 4) [0, 0] See Also ======== prufer_repr: returns Prufer sequence of a Prufer object. rr'N)r intrangeappendpopremove) treendLedgeixyjs rrPrufer.to_prufers(   D 1gJ!OJ 1gJ!OJ  q1uA1X419AQ<QAq'\QA=  HHQKV ttEE!H KK )*rc/n/n[U5S-n[S5nUHnXE==S- ss'M UHSn[U5H nXGS:XdM O XF==S-ss'UW==S-ss'UR[ Xg/55 MU [U5Vs/sHodUS:XdM UPM sn=(d SS/nURU5 U$s snf)aReturn the tree (as a list of edges) of the given Prufer sequence. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> a = Prufer([0, 2], 4) >>> a.tree_repr [[0, 1], [0, 2], [2, 3]] >>> Prufer.to_tree([0, 2]) [[0, 1], [0, 2], [2, 3]] References ========== .. [1] https://hamberg.no/erlend/posts/2010-11-06-prufer-sequence-compact-tree-representation.html See Also ======== tree_repr: returns tree representation of a Prufer object. r,cg)Nr'r>rr Prufer.to_tree..srr'r)lenr r.r/sorted)pruferr2lastr3r4pr7r:s rrPrufer.to_trees0 K!O  "A DAIDA1X419 DAID aDAID KKv '!818atqy81;aV D 2s  C*Cc[5nUSSnUHCn[[U5S- 5H%nX4US-upVXe:aXepeURXV45 M' ME /n[5nS=p)UH\n UR U 5 Ub[ U SU5OU SnU b[ U SU 5OU Sn UR[U 55 M^ [[X)S-55U- n U (aPU Vs/sHoDU-PM n n[U 5S:XaSU R5-n OS[U 5-n [U 5eUS:wa/[U5HupJU V s/sHoU- PM sn Xt'M X-n [U5U S-4$s snfs sn f)aReturn a list of edges and the number of nodes from the given runs that connect nodes in an integer-labelled tree. All node numbers will be shifted so that the minimum node is 0. It is not a problem if edges are repeated in the runs; only unique edges are returned. There is no assumption made about what the range of the node labels should be, but all nodes from the smallest through the largest must be present. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> Prufer.edges([1, 2, 3], [2, 4, 5]) # a T ([[0, 1], [1, 2], [1, 3], [3, 4]], 5) Duplicate edges are removed: >>> Prufer.edges([0, 1, 2, 3], [1, 4, 5], [1, 4, 6]) # a K ([[0, 1], [1, 2], [1, 4], [2, 3], [4, 5], [4, 6]], 7) rr'r,NNode %s is missing.Nodes %s are missing.) setr.rAaddupdateminmaxr/listr0rB ValueError enumerate)runsenminrr7abrvgotnmaxeimissingmsgr3s redges Prufer.edgess0 EAwqzA3q6A:&AE{5qqf ' eB JJrN'+'73r!ud#RUD'+'73r!ud#RUD IId2h   eD(+,s2 )01A4xG17|q +gkkm;-w?S/ ! 19"2+-.2aT2.' LDbz4!8##2/s 2FFcSnSn[URS- SS5H%nXURU-- nX R-nM' U$)zComputes the rank of a Prufer sequence. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> a = Prufer([[0, 1], [0, 2], [0, 3]]) >>> a.prufer_rank() 0 See Also ======== rank, next, prev, size rr')r.rr)rrUrEr7s rr#Prufer.prufer_rank%sS"  tzzA~r2.A 4##A&& &A OA/rc[U5[U5p[[5n[US- SS5HnX-X4'XU- U-nM [ [[ U55Vs/sHoCUPM sn5$s snf)zFinds the unranked Prufer sequence. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> Prufer.unrank(0, 4) Prufer([0, 0]) rarb)r r r-r.r rA)rr$r3r5r7s runrank Prufer.unrank=sv)VD\4  q1ub"%A8ADQ4K!#D&U3q6]3]t]3443s'A<c&US(a[US5O [5nU4[SUSS55-n[R"U/UQ70UD6n[ US5/nUS(a[ USS5(aUSS(d [S5e[U5S:aUSnO[[US55n[U5S-nU[U5:waS[[U55U- n[U5S:XaSUR5-nOS[U5-n[U5eUSV s/sHn [ U 5PM sn UlXTlU$USUl[UR"5S-UlU$s sn f) a>The constructor for the Prufer object. Examples ======== >>> from sympy.combinatorics.prufer import Prufer A Prufer object can be constructed from a list of edges: >>> a = Prufer([[0, 1], [0, 2], [0, 3]]) >>> a.prufer_repr [0, 0] If the number of nodes is given, no checking of the nodes will be performed; it will be assumed that nodes 0 through n - 1 are present: >>> Prufer([[0, 1], [0, 2], [0, 3]], 4) Prufer([[0, 1], [0, 2], [0, 3]], 4) A Prufer object can be constructed from a Prufer sequence: >>> b = Prufer([1, 3]) >>> b.tree_repr [[0, 1], [1, 3], [2, 3]] rc38# UHn[U5v M g7f)Nr).0args r !Prufer.__new__..msAx}}sr'Nz-Prufer expects at least one edge in the tree.rHrIr,)rrtupler__new__rOr rPrArJrrNr.r0rBrrr) clsargskw_argsarg0ret_objnnodesrr\r]r7s rrnPrufer.__new__Psv8"&auT!W~egwAQRAAA--6d6g6T!W  7xQ ++71: CEE4y1}aGDG,-UaSZ'!%-058G7|q(3gkkmC5wG$S/)377!;7a$q'7!;G #N$(7G  !5!56:GN ">> from sympy.combinatorics.prufer import Prufer >>> a = Prufer([[0, 1], [0, 2], [0, 3]]) >>> b = a.next(1) # == a.next() >>> b.tree_repr [[0, 2], [0, 1], [1, 3]] >>> b.rank 1 See Also ======== prufer_rank, rank, prev, size r rer$rrdeltas rnext Prufer.nexts"(}}TYY. ;;rc\[RURU- UR5$)a4Generates the Prufer sequence that is -delta before the current one. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> a = Prufer([[0, 1], [1, 2], [2, 3], [1, 4]]) >>> a.rank 36 >>> b = a.prev() >>> b Prufer([1, 2, 0]) >>> b.rank 35 See Also ======== prufer_rank, rank, next, size rwrxs rr( Prufer.prevs",}}TYY-tzz::r)rr"r)r')__name__ __module__ __qualname____firstlineno____doc__rrrr"propertyrrrr$r) staticmethodrrr^r# classmethodrernrzr(__static_attributes__r>rrr r sLJ F E !!4,  244$00d**X3$3$j055$5n<,;rr N) sympy.corersympy.core.containersrsympy.tensor.arrayrsympy.core.sympifyrsympy.utilities.iterablesrr sympy.utilities.miscr collectionsr r r>rrrs('$'7'#h;Uh;r