4463:
2408:
4479:
4470:
4128:
4439:. In the naive algorithm for agglomerative clustering, implementing a different linkage scheme may be accomplished simply by using a different formula to calculate inter-cluster distances in the algorithm. The formula that should be adjusted has been highlighted using bold text in the above algorithm description. However, more efficient algorithms such as the one described below do not generalize to all linkage schemes in the same way.
4454:
1957:
2403:{\displaystyle {\begin{array}{lllllll}D_{2}((a,b),c)&=&\min(D_{1}(a,c),D_{1}(b,c))&=&\min(21,30)&=&21\\D_{2}((a,b),d)&=&\min(D_{1}(a,d),D_{1}(b,d))&=&\min(31,34)&=&31\\D_{2}((a,b),e)&=&\min(D_{1}(a,e),D_{1}(b,e))&=&\min(23,21)&=&21\end{array}}}
4826:, and this information is sufficient to determine the clustering itself. As Sibson shows, when a new item is added to the set of items, the updated functions representing the new single-linkage clustering for the augmented set, represented in the same way, can be constructed from the old clustering in time
62:
In the beginning of the agglomerative clustering process, each element is in a cluster of its own. The clusters are then sequentially combined into larger clusters, until all elements end up being in the same cluster. At each step, the two clusters separated by the shortest distance are combined. The
70:
In single-linkage clustering, the distance between two clusters is determined by a single pair of elements: those two elements (one in each cluster) that are closest to each other. The shortest of these pairwise distances that remain at any step causes the two clusters whose elements are involved to
45:
This method tends to produce long thin clusters in which nearby elements of the same cluster have small distances, but elements at opposite ends of a cluster may be much farther from each other than two elements of other clusters. For some classes of data, this may lead to difficulties in defining
4116:
4928:
to construct the minimum spanning tree (but not the clustering) of the given items and distances. Then, applying
Kruskal's algorithm to the sparse graph formed by the edges of the minimum spanning tree produces the clustering itself in an additional time
3452:
4858:
An alternative algorithm, running in the same optimal time and space bounds, is based on the equivalence between the naive algorithm and
Kruskal's algorithm for minimum spanning trees. Instead of using Kruskal's algorithm, one can use
3128:
4380:
2985:
4420:. However, in single linkage clustering, the order in which clusters are formed is important, while for minimum spanning trees what matters is the set of pairs of points that form distances chosen by the algorithm.
42:. It is based on grouping clusters in bottom-up fashion (agglomerative clustering), at each step combining two clusters that contain the closest pair of elements not yet belonging to the same cluster as each other.
3900:
3891:
1704:
190:
1808:
2715:
2655:
358:
1555:
1135:
3743:
3673:
3580:
1352:
4965:
259:
4897:
4755:
4586:
4550:
3247:
706:
623:
875:
556:
314:
3487:
3231:
3199:
3172:
2828:
2774:
2742:
2480:
2438:
1949:
1913:
1878:
1851:
1582:
1379:
981:
741:
4994:
4926:
4853:
4824:
4695:
4615:
387:
1007:
949:
923:
793:
767:
582:
467:
441:
2996:
4795:
4775:
4735:
4715:
4675:
4655:
4635:
4402:
4253:
4233:
4213:
4193:
4173:
4153:
3783:
3763:
3693:
3623:
3600:
1287:
1261:
1234:
1207:
1181:
897:
813:
514:
407:
279:
5080:
Feigelson, Eric (2012). "Classification in astronomy: past and present". In Way, Michael J.; Scargle, Jeffrey D.; Ali, Kamal M.; Srivastava, Ashok N. (eds.).
4260:
2872:
4111:{\displaystyle \delta (v,r)=\delta (a,r)-\delta (a,v)=\delta (b,r)-\delta (b,v)=\delta (c,r)-\delta (c,v)=\delta (e,r)-\delta (e,v)=14-10.5=3.5}
5188:
5362:
5284:
5064:
3790:
54:, which may often involve long strings of matter; in this application, it is also known as the friends-of-friends algorithm.
1617:
5279:. Philadelphia, Pa. Alexandria, Va: SIAM, Society for Industrial and Applied Mathematics American Statistical Association.
4127:
2440:
are not affected by the matrix update as they correspond to distances between elements not involved in the first cluster.
108:
1737:
5010:
4491:
4424:
1185:
951:
and adding a row and column corresponding to the newly formed cluster. The proximity between the new cluster, denoted
4855:. The SLINK algorithm then loops over the items, one by one, adding them to the representation of the clustering.
5015:
1150:
232:
39:
4413:
79:, which shows the sequence in which clusters were merged and the distance at which each merge took place.
4462:
4417:
235:
scheme that erases rows and columns in a proximity matrix as old clusters are merged into new ones. The
5115:. Developments in Environmental Modelling. Vol. 20 (Second English ed.). Amsterdam: Elsevier.
4516:
The naive algorithm for single-linkage clustering is easy to understand but slow, with time complexity
2660:
2600:
1962:
5085:
4637:
numbered items by two functions. These functions are both determined by finding the smallest cluster
319:
4797:. Storing these functions in two arrays that map each item number to its function value takes space
4860:
1512:
1012:
5317:
5171:
Olsen GJ (1988). "Phylogenetic analysis using ribosomal RNA". In Noller HF Jr, Moldave K (eds.).
3698:
3628:
3535:
3447:{\displaystyle D_{3}(((a,b),c,e),d)=\min(D_{2}((a,b),d),D_{2}(c,d),D_{2}(e,d))=\min(31,28,43)=28}
1307:
1265:
4932:
238:
5280:
5194:
5184:
5153:
5060:
4866:
4740:
4555:
4519:
1159:
628:
590:
5309:
5255:
5240:
5221:
5176:
5143:
5135:
5093:
5025:
5005:
4436:
1154:
818:
526:
284:
5347:
5329:
3465:
3204:
3177:
3150:
3123:{\displaystyle \delta (u,v)=\delta (c,v)-\delta (a,u)=\delta (c,v)-\delta (b,u)=10.5-8.5=2}
2801:
2747:
2720:
2458:
2416:
1922:
1891:
1856:
1829:
1560:
1357:
954:
714:
5325:
5020:
4970:
4902:
4829:
4800:
4680:
4591:
4478:
4469:
363:
986:
928:
902:
772:
746:
561:
446:
420:
17:
5300:
Gower JC, Ross GJ (1969). "Minimum spanning trees and single linkage cluster analysis".
5089:
4780:
4760:
4720:
4700:
4660:
4640:
4620:
4617:(both optimal) known as SLINK. The slink algorithm represents a clustering on a set of
4387:
4238:
4218:
4198:
4178:
4158:
4138:
3768:
3748:
3678:
3608:
3585:
1719:
1272:
1246:
1219:
1192:
1166:
882:
798:
472:
392:
264:
51:
5212:
Murtagh F, Contreras P (2012). "Algorithms for hierarchical clustering: an overview".
5180:
5148:
5127:
3201:(see below), reduced in size by two rows and two columns because of the clustering of
5356:
4443:
Comparison of dendrograms obtained under different clustering methods from the same
1880:(see below), reduced in size by one row and one column because of the clustering of
4375:{\displaystyle \delta (a,r)=\delta (b,r)=\delta (c,r)=\delta (e,r)=\delta (d,r)=14}
2455:
We now reiterate the three previous actions, starting from the new distance matrix
1238:
2838:
are now connected. Because of the ultrametricity constraint, the branches joining
4453:
5139:
76:
31:
5260:
4412:
The naive algorithm for single linkage clustering is essentially the same as
2980:{\displaystyle \delta (a,v)=\delta (b,v)=\delta (c,v)=\delta (e,v)=21/2=10.5}
5241:"SLINK: an optimally efficient algorithm for the single-link cluster method"
1211:
47:
587:
Find the most similar pair of clusters in the current clustering, say pair
63:
function used to determine the distance between two clusters, known as the
5198:
5157:
708:
where the minimum is over all pairs of clusters in the current clustering.
46:
classes that could usefully subdivide the data. However, it is popular in
5321:
5097:
5313:
5225:
5214:
4135:
The dendrogram is now complete. It is ultrametric because all tips (
5128:"Collection of published 5S, 5.8S and 4.5S ribosomal RNA sequences"
5035:
5030:
4503:
4497:
4432:
4428:
519:
The single linkage algorithm is composed of the following steps:
4552:. In 1973, R. Sibson proposed an algorithm with time complexity
67:, is what differentiates the agglomerative clustering methods.
1915:
correspond to the new distances, calculated by retaining the
899:, by deleting the rows and columns corresponding to clusters
4477:
4468:
4452:
4126:
1140:
If all objects are in one cluster, stop. Else, go to step 2.
5277:
Data clustering : theory, algorithms, and applications
4677:
and at least one larger-numbered item. The first function,
3886:{\displaystyle \delta (((a,b),c,e),r)=\delta (d,r)=28/2=14}
5175:. Methods in Enzymology. Vol. 164. pp. 793–812.
5082:
Advances in
Machine Learning and Data Mining for Astronomy
203:
are any two sets of elements considered as clusters, and
4777:
to the distance associated with the creation of cluster
1826:
We then proceed to update the initial proximity matrix
4863:, in a variation without binary heaps that takes time
1699:{\displaystyle \delta (a,u)=\delta (b,u)=D_{1}(a,b)/2}
75:. The result of the clustering can be visualized as a
4973:
4935:
4905:
4869:
4832:
4803:
4783:
4763:
4743:
4723:
4703:
4683:
4663:
4643:
4623:
4594:
4558:
4522:
4390:
4263:
4241:
4221:
4201:
4181:
4161:
4141:
3903:
3793:
3771:
3751:
3701:
3681:
3631:
3611:
3588:
3538:
3468:
3250:
3207:
3180:
3153:
2999:
2875:
2804:
2750:
2723:
2663:
2603:
2461:
2419:
1960:
1925:
1894:
1859:
1832:
1740:
1620:
1563:
1515:
1360:
1310:
1275:
1249:
1222:
1195:
1169:
1015:
989:
957:
931:
905:
885:
821:
801:
775:
749:
717:
631:
593:
564:
529:
475:
449:
423:
395:
366:
322:
287:
267:
241:
111:
82:
Mathematically, the linkage function – the distance
185:{\displaystyle D(X,Y)=\min _{x\in X,y\in Y}d(x,y),}
5302:Journal of the Royal Statistical Society, Series C
4988:
4959:
4920:
4891:
4847:
4818:
4789:
4769:
4749:
4729:
4709:
4689:
4669:
4649:
4629:
4609:
4580:
4544:
4396:
4374:
4247:
4227:
4207:
4187:
4167:
4147:
4110:
3885:
3777:
3757:
3737:
3687:
3667:
3617:
3594:
3574:
3481:
3446:
3225:
3193:
3166:
3122:
2979:
2822:
2768:
2736:
2709:
2649:
2474:
2432:
2402:
1943:
1907:
1872:
1845:
1803:{\displaystyle \delta (a,u)=\delta (b,u)=17/2=8.5}
1802:
1698:
1576:
1549:
1373:
1346:
1281:
1255:
1228:
1201:
1175:
1129:
1001:
975:
943:
917:
891:
869:
807:
795:into a single cluster to form the next clustering
787:
761:
735:
700:
617:
576:
550:
508:
461:
435:
401:
381:
352:
308:
273:
253:
184:
523:Begin with the disjoint clustering having level
3414:
3309:
2368:
2299:
2223:
2154:
2078:
2009:
1055:
665:
316:. The clusterings are assigned sequence numbers
215:) denotes the distance between the two elements
134:
2866:are equal and have the following total length:
409:-th clustering. A cluster with sequence number
1718:. This corresponds to the expectation of the
8:
1124:
1058:
58:Overview of agglomerative clustering methods
27:Agglomerative hierarchical clustering method
1919:between each element of the first cluster
1153:genetic distance matrix computed from the
5259:
5147:
4972:
4934:
4904:
4880:
4868:
4831:
4802:
4782:
4762:
4742:
4722:
4702:
4682:
4662:
4642:
4622:
4593:
4569:
4557:
4533:
4521:
4389:
4262:
4240:
4220:
4200:
4180:
4160:
4140:
3902:
3869:
3792:
3770:
3750:
3700:
3680:
3630:
3610:
3587:
3537:
3473:
3467:
3387:
3359:
3319:
3255:
3249:
3206:
3185:
3179:
3158:
3152:
2998:
2963:
2874:
2803:
2749:
2728:
2722:
2668:
2662:
2608:
2602:
2466:
2460:
2424:
2418:
2337:
2309:
2259:
2192:
2164:
2114:
2047:
2019:
1969:
1961:
1959:
1924:
1899:
1893:
1864:
1858:
1837:
1831:
1786:
1739:
1688:
1667:
1619:
1568:
1562:
1520:
1514:
1365:
1359:
1309:
1304:Let us assume that we have five elements
1274:
1248:
1221:
1194:
1168:
1014:
988:
956:
930:
904:
884:
820:
800:
774:
748:
716:
630:
592:
563:
528:
474:
448:
422:
394:
365:
321:
286:
266:
240:
137:
110:
4717:to the largest-numbered item in cluster
4441:
3695:are now connected. The branches joining
3491:
2484:
1385:
5084:. Chapman and Hall/CRC. pp. 3–10.
5059:. Chichester, West Sussex, U.K: Wiley.
5047:
3895:We deduce the remaining branch length:
71:be merged. The method is also known as
5254:(1). British Computer Society: 30–34.
4384:The dendrogram is therefore rooted by
815:. Set the level of this clustering to
4444:
2990:We deduce the missing branch length:
1157:sequence alignment of five bacteria:
417:) and the proximity between clusters
7:
4423:Alternative linkage schemes include
1951:and each of the remaining elements:
1381:of pairwise distances between them:
1149:This working example is based on a
5220:(1). Wiley Online Library: 86–97.
3174:matrix into a new distance matrix
25:
4131:Single Linkage Dendrogram 5S data
2710:{\displaystyle D_{2}((a,b),e)=21}
2650:{\displaystyle D_{2}((a,b),c)=21}
1722:hypothesis. The branches joining
102:– is described by the expression
4461:
3625:denote the (root) node to which
3132:
1812:
5111:Legendre P, Legendre L (1998).
2789:Second branch length estimation
711:Increment the sequence number:
353:{\displaystyle 0,1,\ldots ,n-1}
5126:Erdmann VA, Wolters J (1986).
4983:
4977:
4954:
4939:
4915:
4909:
4886:
4873:
4842:
4836:
4813:
4807:
4604:
4598:
4575:
4562:
4539:
4526:
4427:, average linkage clustering (
4363:
4351:
4342:
4330:
4321:
4309:
4300:
4288:
4279:
4267:
4087:
4075:
4066:
4054:
4045:
4033:
4024:
4012:
4003:
3991:
3982:
3970:
3961:
3949:
3940:
3928:
3919:
3907:
3860:
3848:
3839:
3830:
3815:
3803:
3800:
3797:
3732:
3717:
3705:
3702:
3662:
3647:
3635:
3632:
3569:
3554:
3542:
3539:
3435:
3417:
3408:
3405:
3393:
3377:
3365:
3349:
3340:
3328:
3325:
3312:
3303:
3294:
3279:
3267:
3264:
3261:
3220:
3208:
3147:We then proceed to update the
3099:
3087:
3078:
3066:
3057:
3045:
3036:
3024:
3015:
3003:
2954:
2942:
2933:
2921:
2912:
2900:
2891:
2879:
2817:
2805:
2763:
2751:
2698:
2689:
2677:
2674:
2638:
2629:
2617:
2614:
2383:
2371:
2358:
2355:
2343:
2327:
2315:
2302:
2289:
2280:
2268:
2265:
2238:
2226:
2213:
2210:
2198:
2182:
2170:
2157:
2144:
2135:
2123:
2120:
2093:
2081:
2068:
2065:
2053:
2037:
2025:
2012:
1999:
1990:
1978:
1975:
1938:
1926:
1777:
1765:
1756:
1744:
1685:
1673:
1657:
1645:
1636:
1624:
1597:First branch length estimation
1538:
1526:
1341:
1311:
1121:
1118:
1112:
1106:
1100:
1097:
1088:
1085:
1079:
1073:
1067:
1064:
1049:
1046:
1040:
1034:
1022:
1019:
996:
990:
970:
958:
938:
932:
912:
906:
864:
861:
855:
849:
843:
840:
831:
825:
782:
776:
756:
750:
695:
692:
686:
680:
674:
671:
659:
656:
650:
644:
638:
635:
612:
606:
600:
594:
539:
533:
503:
500:
494:
488:
482:
479:
456:
450:
430:
424:
376:
370:
303:
291:
231:The following algorithm is an
176:
164:
127:
115:
1:
5181:10.1016/s0076-6879(88)64084-5
4657:that contains both item
4121:The single-linkage dendrogram
3142:Second distance matrix update
1550:{\displaystyle D_{1}(a,b)=17}
1130:{\displaystyle d=\min\{d,d\}}
879:Update the proximity matrix,
38:is one of several methods of
4502:Average linkage clustering:
4496:Average linkage clustering:
1853:into a new proximity matrix
1821:First distance matrix update
1614:are now connected. Setting
73:nearest neighbour clustering
5363:Cluster analysis algorithms
5134:. 14 Suppl (Suppl): r1-59.
5011:Complete-linkage clustering
4492:Complete-linkage clustering
4425:complete linkage clustering
3738:{\displaystyle ((a,b),c,e)}
3668:{\displaystyle ((a,b),c,e)}
3575:{\displaystyle ((a,b),c,e)}
1347:{\displaystyle (a,b,c,d,e)}
1186:Bacillus stearothermophilus
5379:
4960:{\displaystyle O(n\log n)}
4488:Single-linkage clustering
2798:denote the node to which
2717:are the lowest values of
1606:denote the node to which
1584:, so we cluster elements
1354:and the following matrix
254:{\displaystyle N\times N}
36:single-linkage clustering
18:Single linkage clustering
4892:{\displaystyle O(n^{2})}
4750:{\displaystyle \lambda }
4581:{\displaystyle O(n^{2})}
4545:{\displaystyle O(n^{3})}
3133:see the final dendrogram
1813:see the final dendrogram
701:{\displaystyle d=\min d}
5348:Linkages used in Matlab
5140:10.1093/nar/14.suppl.r1
5016:Hierarchical clustering
4737:. The second function,
4235:) are equidistant from
1557:is the lowest value of
618:{\displaystyle (r),(s)}
281:contains all distances
40:hierarchical clustering
5261:10.1093/comjnl/16.1.30
5132:Nucleic Acids Research
4990:
4961:
4922:
4893:
4849:
4820:
4791:
4771:
4751:
4731:
4711:
4691:
4671:
4651:
4631:
4611:
4582:
4546:
4482:
4473:
4457:
4418:minimum spanning trees
4398:
4376:
4249:
4229:
4209:
4189:
4169:
4149:
4132:
4112:
3887:
3779:
3759:
3739:
3689:
3669:
3619:
3596:
3576:
3483:
3448:
3227:
3195:
3168:
3124:
2981:
2824:
2770:
2738:
2711:
2651:
2476:
2434:
2404:
1945:
1909:
1874:
1847:
1804:
1706:ensures that elements
1700:
1578:
1551:
1375:
1348:
1283:
1257:
1230:
1203:
1177:
1131:
1003:
977:
945:
919:
893:
871:
870:{\displaystyle L(m)=d}
809:
789:
763:
737:
702:
619:
578:
552:
551:{\displaystyle L(0)=0}
510:
463:
437:
403:
383:
354:
310:
309:{\displaystyle d(i,j)}
275:
255:
186:
4991:
4962:
4923:
4894:
4850:
4821:
4792:
4772:
4752:
4732:
4712:
4692:
4672:
4652:
4632:
4612:
4588:and space complexity
4583:
4547:
4481:
4472:
4456:
4399:
4377:
4250:
4230:
4210:
4190:
4170:
4150:
4130:
4113:
3888:
3780:
3760:
3740:
3690:
3670:
3620:
3597:
3577:
3484:
3482:{\displaystyle D_{3}}
3449:
3228:
3226:{\displaystyle (a,b)}
3196:
3194:{\displaystyle D_{3}}
3169:
3167:{\displaystyle D_{2}}
3125:
2982:
2825:
2823:{\displaystyle (a,b)}
2771:
2769:{\displaystyle (a,b)}
2744:, so we join cluster
2739:
2737:{\displaystyle D_{2}}
2712:
2652:
2477:
2475:{\displaystyle D_{2}}
2435:
2433:{\displaystyle D_{2}}
2413:Italicized values in
2405:
1946:
1944:{\displaystyle (a,b)}
1910:
1908:{\displaystyle D_{2}}
1875:
1873:{\displaystyle D_{2}}
1848:
1846:{\displaystyle D_{1}}
1805:
1714:are equidistant from
1701:
1579:
1577:{\displaystyle D_{1}}
1552:
1376:
1374:{\displaystyle D_{1}}
1349:
1284:
1258:
1231:
1204:
1178:
1132:
1004:
978:
976:{\displaystyle (r,s)}
946:
920:
894:
872:
810:
790:
764:
738:
736:{\displaystyle m=m+1}
703:
620:
579:
553:
511:
464:
438:
404:
384:
355:
311:
276:
256:
187:
5248:The Computer Journal
4989:{\displaystyle O(n)}
4971:
4933:
4921:{\displaystyle O(n)}
4903:
4867:
4848:{\displaystyle O(n)}
4830:
4819:{\displaystyle O(n)}
4801:
4781:
4761:
4741:
4721:
4701:
4690:{\displaystyle \pi }
4681:
4661:
4641:
4621:
4610:{\displaystyle O(n)}
4592:
4556:
4520:
4404:, its deepest node.
4388:
4261:
4239:
4219:
4199:
4179:
4159:
4139:
3901:
3791:
3769:
3749:
3699:
3679:
3629:
3609:
3586:
3536:
3532:So we join clusters
3466:
3248:
3205:
3178:
3151:
2997:
2873:
2802:
2748:
2721:
2661:
2601:
2459:
2417:
1958:
1923:
1892:
1857:
1830:
1738:
1618:
1561:
1513:
1358:
1308:
1273:
1247:
1220:
1193:
1167:
1013:
987:
955:
929:
903:
883:
819:
799:
773:
747:
715:
629:
591:
562:
558:and sequence number
527:
473:
447:
421:
393:
389:is the level of the
382:{\displaystyle L(k)}
364:
320:
285:
265:
239:
109:
5090:2012amld.book....3F
4448:
4414:Kruskal's algorithm
3785:then have lengths:
1734:then have lengths
1002:{\displaystyle (k)}
983:and an old cluster
944:{\displaystyle (s)}
918:{\displaystyle (r)}
788:{\displaystyle (s)}
762:{\displaystyle (r)}
577:{\displaystyle m=0}
462:{\displaystyle (s)}
436:{\displaystyle (r)}
94:) between clusters
5055:Everitt B (2011).
4986:
4957:
4918:
4889:
4845:
4816:
4787:
4767:
4747:
4727:
4707:
4687:
4667:
4647:
4627:
4607:
4578:
4542:
4483:
4474:
4458:
4442:
4394:
4372:
4245:
4225:
4205:
4185:
4165:
4145:
4133:
4108:
3883:
3775:
3755:
3735:
3685:
3665:
3615:
3592:
3572:
3479:
3444:
3223:
3191:
3164:
3120:
2977:
2820:
2766:
2734:
2707:
2647:
2472:
2430:
2400:
2398:
1941:
1905:
1870:
1843:
1800:
1696:
1574:
1547:
1371:
1344:
1279:
1266:Micrococcus luteus
1253:
1226:
1199:
1173:
1127:
999:
973:
941:
915:
889:
867:
805:
785:
759:
733:
698:
615:
574:
548:
506:
459:
433:
399:
379:
350:
306:
271:
251:
182:
160:
5239:Sibson R (1973).
5190:978-0-12-182065-7
5113:Numerical Ecology
4790:{\displaystyle C}
4770:{\displaystyle i}
4757:, maps item
4730:{\displaystyle C}
4710:{\displaystyle i}
4697:, maps item
4670:{\displaystyle i}
4650:{\displaystyle C}
4630:{\displaystyle n}
4512:Faster algorithms
4509:
4508:
4397:{\displaystyle r}
4248:{\displaystyle r}
4228:{\displaystyle d}
4208:{\displaystyle e}
4188:{\displaystyle c}
4168:{\displaystyle b}
4148:{\displaystyle a}
3778:{\displaystyle r}
3758:{\displaystyle d}
3688:{\displaystyle d}
3618:{\displaystyle r}
3595:{\displaystyle d}
3530:
3529:
2780:and with element
2595:
2594:
2450:Second clustering
1888:. Bold values in
1509:In this example,
1507:
1506:
1282:{\displaystyle e}
1256:{\displaystyle d}
1229:{\displaystyle c}
1202:{\displaystyle b}
1176:{\displaystyle a}
1160:Bacillus subtilis
892:{\displaystyle D}
808:{\displaystyle m}
743:. Merge clusters
509:{\displaystyle d}
402:{\displaystyle k}
274:{\displaystyle D}
261:proximity matrix
133:
16:(Redirected from
5370:
5335:
5333:
5297:
5291:
5290:
5272:
5266:
5265:
5263:
5245:
5236:
5230:
5229:
5209:
5203:
5202:
5168:
5162:
5161:
5151:
5123:
5117:
5116:
5108:
5102:
5101:
5098:10.1201/b11822-7
5077:
5071:
5070:
5057:Cluster analysis
5052:
5026:Neighbor-joining
5006:Cluster analysis
4995:
4993:
4992:
4987:
4966:
4964:
4963:
4958:
4927:
4925:
4924:
4919:
4898:
4896:
4895:
4890:
4885:
4884:
4861:Prim's algorithm
4854:
4852:
4851:
4846:
4825:
4823:
4822:
4817:
4796:
4794:
4793:
4788:
4776:
4774:
4773:
4768:
4756:
4754:
4753:
4748:
4736:
4734:
4733:
4728:
4716:
4714:
4713:
4708:
4696:
4694:
4693:
4688:
4676:
4674:
4673:
4668:
4656:
4654:
4653:
4648:
4636:
4634:
4633:
4628:
4616:
4614:
4613:
4608:
4587:
4585:
4584:
4579:
4574:
4573:
4551:
4549:
4548:
4543:
4538:
4537:
4465:
4449:
4403:
4401:
4400:
4395:
4381:
4379:
4378:
4373:
4254:
4252:
4251:
4246:
4234:
4232:
4231:
4226:
4214:
4212:
4211:
4206:
4194:
4192:
4191:
4186:
4174:
4172:
4171:
4166:
4154:
4152:
4151:
4146:
4117:
4115:
4114:
4109:
3892:
3890:
3889:
3884:
3873:
3784:
3782:
3781:
3776:
3764:
3762:
3761:
3756:
3744:
3742:
3741:
3736:
3694:
3692:
3691:
3686:
3674:
3672:
3671:
3666:
3624:
3622:
3621:
3616:
3601:
3599:
3598:
3593:
3581:
3579:
3578:
3573:
3492:
3488:
3486:
3485:
3480:
3478:
3477:
3453:
3451:
3450:
3445:
3392:
3391:
3364:
3363:
3324:
3323:
3260:
3259:
3240:
3236:
3232:
3230:
3229:
3224:
3200:
3198:
3197:
3192:
3190:
3189:
3173:
3171:
3170:
3165:
3163:
3162:
3129:
3127:
3126:
3121:
2986:
2984:
2983:
2978:
2967:
2865:
2861:
2857:
2853:
2849:
2845:
2841:
2837:
2833:
2829:
2827:
2826:
2821:
2797:
2783:
2779:
2775:
2773:
2772:
2767:
2743:
2741:
2740:
2735:
2733:
2732:
2716:
2714:
2713:
2708:
2673:
2672:
2656:
2654:
2653:
2648:
2613:
2612:
2485:
2481:
2479:
2478:
2473:
2471:
2470:
2439:
2437:
2436:
2431:
2429:
2428:
2409:
2407:
2406:
2401:
2399:
2342:
2341:
2314:
2313:
2264:
2263:
2197:
2196:
2169:
2168:
2119:
2118:
2052:
2051:
2024:
2023:
1974:
1973:
1950:
1948:
1947:
1942:
1917:minimum distance
1914:
1912:
1911:
1906:
1904:
1903:
1887:
1883:
1879:
1877:
1876:
1871:
1869:
1868:
1852:
1850:
1849:
1844:
1842:
1841:
1809:
1807:
1806:
1801:
1790:
1733:
1729:
1725:
1717:
1713:
1709:
1705:
1703:
1702:
1697:
1692:
1672:
1671:
1613:
1609:
1605:
1591:
1587:
1583:
1581:
1580:
1575:
1573:
1572:
1556:
1554:
1553:
1548:
1525:
1524:
1386:
1380:
1378:
1377:
1372:
1370:
1369:
1353:
1351:
1350:
1345:
1299:First clustering
1288:
1286:
1285:
1280:
1262:
1260:
1259:
1254:
1235:
1233:
1232:
1227:
1208:
1206:
1205:
1200:
1182:
1180:
1179:
1174:
1155:5S ribosomal RNA
1136:
1134:
1133:
1128:
1008:
1006:
1005:
1000:
982:
980:
979:
974:
950:
948:
947:
942:
924:
922:
921:
916:
898:
896:
895:
890:
876:
874:
873:
868:
814:
812:
811:
806:
794:
792:
791:
786:
768:
766:
765:
760:
742:
740:
739:
734:
707:
705:
704:
699:
624:
622:
621:
616:
583:
581:
580:
575:
557:
555:
554:
549:
515:
513:
512:
507:
468:
466:
465:
460:
442:
440:
439:
434:
408:
406:
405:
400:
388:
386:
385:
380:
359:
357:
356:
351:
315:
313:
312:
307:
280:
278:
277:
272:
260:
258:
257:
252:
191:
189:
188:
183:
159:
65:linkage function
21:
5378:
5377:
5373:
5372:
5371:
5369:
5368:
5367:
5353:
5352:
5344:
5339:
5338:
5314:10.2307/2346439
5299:
5298:
5294:
5287:
5274:
5273:
5269:
5243:
5238:
5237:
5233:
5226:10.1002/widm.53
5211:
5210:
5206:
5191:
5170:
5169:
5165:
5125:
5124:
5120:
5110:
5109:
5105:
5079:
5078:
5074:
5067:
5054:
5053:
5049:
5044:
5021:Molecular clock
5002:
4969:
4968:
4931:
4930:
4901:
4900:
4876:
4865:
4864:
4828:
4827:
4799:
4798:
4779:
4778:
4759:
4758:
4739:
4738:
4719:
4718:
4699:
4698:
4679:
4678:
4659:
4658:
4639:
4638:
4619:
4618:
4590:
4589:
4565:
4554:
4553:
4529:
4518:
4517:
4514:
4445:distance matrix
4410:
4386:
4385:
4259:
4258:
4237:
4236:
4217:
4216:
4197:
4196:
4177:
4176:
4157:
4156:
4137:
4136:
4123:
3899:
3898:
3789:
3788:
3767:
3766:
3747:
3746:
3697:
3696:
3677:
3676:
3627:
3626:
3607:
3606:
3584:
3583:
3534:
3533:
3469:
3464:
3463:
3460:
3383:
3355:
3315:
3251:
3246:
3245:
3238:
3234:
3203:
3202:
3181:
3176:
3175:
3154:
3149:
3148:
2995:
2994:
2871:
2870:
2863:
2859:
2855:
2851:
2847:
2843:
2839:
2835:
2831:
2800:
2799:
2795:
2781:
2777:
2746:
2745:
2724:
2719:
2718:
2664:
2659:
2658:
2604:
2599:
2598:
2462:
2457:
2456:
2446:
2420:
2415:
2414:
2397:
2396:
2391:
2386:
2366:
2361:
2333:
2305:
2297:
2292:
2255:
2252:
2251:
2246:
2241:
2221:
2216:
2188:
2160:
2152:
2147:
2110:
2107:
2106:
2101:
2096:
2076:
2071:
2043:
2015:
2007:
2002:
1965:
1956:
1955:
1921:
1920:
1895:
1890:
1889:
1885:
1881:
1860:
1855:
1854:
1833:
1828:
1827:
1736:
1735:
1731:
1727:
1723:
1715:
1711:
1707:
1663:
1616:
1615:
1611:
1607:
1603:
1589:
1585:
1564:
1559:
1558:
1516:
1511:
1510:
1361:
1356:
1355:
1306:
1305:
1295:
1271:
1270:
1245:
1244:
1218:
1217:
1191:
1190:
1165:
1164:
1147:
1145:Working example
1011:
1010:
985:
984:
953:
952:
927:
926:
901:
900:
881:
880:
817:
816:
797:
796:
771:
770:
745:
744:
713:
712:
627:
626:
625:, according to
589:
588:
560:
559:
525:
524:
471:
470:
445:
444:
419:
418:
391:
390:
362:
361:
318:
317:
283:
282:
263:
262:
237:
236:
229:
227:Naive algorithm
107:
106:
60:
52:galaxy clusters
28:
23:
22:
15:
12:
11:
5:
5376:
5374:
5366:
5365:
5355:
5354:
5351:
5350:
5343:
5342:External links
5340:
5337:
5336:
5292:
5285:
5275:Gan G (2007).
5267:
5231:
5204:
5189:
5163:
5118:
5103:
5072:
5065:
5046:
5045:
5043:
5040:
5039:
5038:
5033:
5028:
5023:
5018:
5013:
5008:
5001:
4998:
4985:
4982:
4979:
4976:
4956:
4953:
4950:
4947:
4944:
4941:
4938:
4917:
4914:
4911:
4908:
4888:
4883:
4879:
4875:
4872:
4844:
4841:
4838:
4835:
4815:
4812:
4809:
4806:
4786:
4766:
4746:
4726:
4706:
4686:
4666:
4646:
4626:
4606:
4603:
4600:
4597:
4577:
4572:
4568:
4564:
4561:
4541:
4536:
4532:
4528:
4525:
4513:
4510:
4507:
4506:
4500:
4494:
4489:
4485:
4484:
4475:
4466:
4459:
4409:
4408:Other linkages
4406:
4393:
4371:
4368:
4365:
4362:
4359:
4356:
4353:
4350:
4347:
4344:
4341:
4338:
4335:
4332:
4329:
4326:
4323:
4320:
4317:
4314:
4311:
4308:
4305:
4302:
4299:
4296:
4293:
4290:
4287:
4284:
4281:
4278:
4275:
4272:
4269:
4266:
4244:
4224:
4204:
4184:
4164:
4144:
4122:
4119:
4107:
4104:
4101:
4098:
4095:
4092:
4089:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4032:
4029:
4026:
4023:
4020:
4017:
4014:
4011:
4008:
4005:
4002:
3999:
3996:
3993:
3990:
3987:
3984:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3960:
3957:
3954:
3951:
3948:
3945:
3942:
3939:
3936:
3933:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3909:
3906:
3882:
3879:
3876:
3872:
3868:
3865:
3862:
3859:
3856:
3853:
3850:
3847:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3823:
3820:
3817:
3814:
3811:
3808:
3805:
3802:
3799:
3796:
3774:
3754:
3734:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3684:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3614:
3591:
3571:
3568:
3565:
3562:
3559:
3556:
3553:
3550:
3547:
3544:
3541:
3528:
3527:
3524:
3519:
3515:
3514:
3509:
3506:
3502:
3501:
3498:
3495:
3476:
3472:
3459:
3456:
3455:
3454:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3398:
3395:
3390:
3386:
3382:
3379:
3376:
3373:
3370:
3367:
3362:
3358:
3354:
3351:
3348:
3345:
3342:
3339:
3336:
3333:
3330:
3327:
3322:
3318:
3314:
3311:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3258:
3254:
3222:
3219:
3216:
3213:
3210:
3188:
3184:
3161:
3157:
3145:
3144:
3138:
3137:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
3020:
3017:
3014:
3011:
3008:
3005:
3002:
2988:
2987:
2976:
2973:
2970:
2966:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2819:
2816:
2813:
2810:
2807:
2792:
2791:
2765:
2762:
2759:
2756:
2753:
2731:
2727:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2682:
2679:
2676:
2671:
2667:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2625:
2622:
2619:
2616:
2611:
2607:
2593:
2592:
2589:
2584:
2579:
2574:
2570:
2569:
2564:
2561:
2556:
2551:
2547:
2546:
2541:
2536:
2533:
2528:
2524:
2523:
2518:
2513:
2508:
2505:
2501:
2500:
2497:
2494:
2491:
2488:
2469:
2465:
2453:
2452:
2445:
2442:
2427:
2423:
2411:
2410:
2395:
2392:
2390:
2387:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2365:
2362:
2360:
2357:
2354:
2351:
2348:
2345:
2340:
2336:
2332:
2329:
2326:
2323:
2320:
2317:
2312:
2308:
2304:
2301:
2298:
2296:
2293:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2262:
2258:
2254:
2253:
2250:
2247:
2245:
2242:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2220:
2217:
2215:
2212:
2209:
2206:
2203:
2200:
2195:
2191:
2187:
2184:
2181:
2178:
2175:
2172:
2167:
2163:
2159:
2156:
2153:
2151:
2148:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2117:
2113:
2109:
2108:
2105:
2102:
2100:
2097:
2095:
2092:
2089:
2086:
2083:
2080:
2077:
2075:
2072:
2070:
2067:
2064:
2061:
2058:
2055:
2050:
2046:
2042:
2039:
2036:
2033:
2030:
2027:
2022:
2018:
2014:
2011:
2008:
2006:
2003:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1972:
1968:
1964:
1963:
1940:
1937:
1934:
1931:
1928:
1902:
1898:
1867:
1863:
1840:
1836:
1824:
1823:
1799:
1796:
1793:
1789:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1720:ultrametricity
1695:
1691:
1687:
1684:
1681:
1678:
1675:
1670:
1666:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1600:
1599:
1571:
1567:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1523:
1519:
1505:
1504:
1501:
1498:
1495:
1492:
1489:
1485:
1484:
1481:
1478:
1475:
1472:
1469:
1465:
1464:
1461:
1458:
1455:
1452:
1449:
1445:
1444:
1441:
1438:
1435:
1432:
1429:
1425:
1424:
1421:
1418:
1415:
1412:
1409:
1405:
1404:
1401:
1398:
1395:
1392:
1389:
1368:
1364:
1343:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1302:
1301:
1294:
1291:
1278:
1252:
1225:
1198:
1172:
1146:
1143:
1142:
1141:
1138:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1009:is defined as
998:
995:
992:
972:
969:
966:
963:
960:
940:
937:
934:
914:
911:
908:
888:
877:
866:
863:
860:
857:
854:
851:
848:
845:
842:
839:
836:
833:
830:
827:
824:
804:
784:
781:
778:
758:
755:
752:
732:
729:
726:
723:
720:
709:
697:
694:
691:
688:
685:
682:
679:
676:
673:
670:
667:
664:
661:
658:
655:
652:
649:
646:
643:
640:
637:
634:
614:
611:
608:
605:
602:
599:
596:
585:
573:
570:
567:
547:
544:
541:
538:
535:
532:
505:
502:
499:
496:
493:
490:
487:
484:
481:
478:
458:
455:
452:
432:
429:
426:
398:
378:
375:
372:
369:
349:
346:
343:
340:
337:
334:
331:
328:
325:
305:
302:
299:
296:
293:
290:
270:
250:
247:
244:
228:
225:
193:
192:
181:
178:
175:
172:
169:
166:
163:
158:
155:
152:
149:
146:
143:
140:
136:
132:
129:
126:
123:
120:
117:
114:
59:
56:
50:for analyzing
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5375:
5364:
5361:
5360:
5358:
5349:
5346:
5345:
5341:
5331:
5327:
5323:
5319:
5315:
5311:
5307:
5303:
5296:
5293:
5288:
5286:9780898716238
5282:
5278:
5271:
5268:
5262:
5257:
5253:
5249:
5242:
5235:
5232:
5227:
5223:
5219:
5215:
5208:
5205:
5200:
5196:
5192:
5186:
5182:
5178:
5174:
5167:
5164:
5159:
5155:
5150:
5145:
5141:
5137:
5133:
5129:
5122:
5119:
5114:
5107:
5104:
5099:
5095:
5091:
5087:
5083:
5076:
5073:
5068:
5066:9780470749913
5062:
5058:
5051:
5048:
5041:
5037:
5034:
5032:
5029:
5027:
5024:
5022:
5019:
5017:
5014:
5012:
5009:
5007:
5004:
5003:
4999:
4997:
4980:
4974:
4951:
4948:
4945:
4942:
4936:
4912:
4906:
4881:
4877:
4870:
4862:
4856:
4839:
4833:
4810:
4804:
4784:
4764:
4744:
4724:
4704:
4684:
4664:
4644:
4624:
4601:
4595:
4570:
4566:
4559:
4534:
4530:
4523:
4511:
4505:
4501:
4499:
4495:
4493:
4490:
4487:
4486:
4480:
4476:
4471:
4467:
4464:
4460:
4455:
4451:
4450:
4446:
4440:
4438:
4437:Ward's method
4434:
4430:
4426:
4421:
4419:
4415:
4407:
4405:
4391:
4382:
4369:
4366:
4360:
4357:
4354:
4348:
4345:
4339:
4336:
4333:
4327:
4324:
4318:
4315:
4312:
4306:
4303:
4297:
4294:
4291:
4285:
4282:
4276:
4273:
4270:
4264:
4256:
4242:
4222:
4202:
4182:
4162:
4142:
4129:
4125:
4120:
4118:
4105:
4102:
4099:
4096:
4093:
4090:
4084:
4081:
4078:
4072:
4069:
4063:
4060:
4057:
4051:
4048:
4042:
4039:
4036:
4030:
4027:
4021:
4018:
4015:
4009:
4006:
4000:
3997:
3994:
3988:
3985:
3979:
3976:
3973:
3967:
3964:
3958:
3955:
3952:
3946:
3943:
3937:
3934:
3931:
3925:
3922:
3916:
3913:
3910:
3904:
3896:
3893:
3880:
3877:
3874:
3870:
3866:
3863:
3857:
3854:
3851:
3845:
3842:
3836:
3833:
3827:
3824:
3821:
3818:
3812:
3809:
3806:
3794:
3786:
3772:
3752:
3729:
3726:
3723:
3720:
3714:
3711:
3708:
3682:
3659:
3656:
3653:
3650:
3644:
3641:
3638:
3612:
3603:
3589:
3566:
3563:
3560:
3557:
3551:
3548:
3545:
3525:
3523:
3520:
3517:
3516:
3513:
3510:
3507:
3504:
3503:
3499:
3496:
3494:
3493:
3490:
3474:
3470:
3457:
3441:
3438:
3432:
3429:
3426:
3423:
3420:
3411:
3402:
3399:
3396:
3388:
3384:
3380:
3374:
3371:
3368:
3360:
3356:
3352:
3346:
3343:
3337:
3334:
3331:
3320:
3316:
3306:
3300:
3297:
3291:
3288:
3285:
3282:
3276:
3273:
3270:
3256:
3252:
3244:
3243:
3242:
3217:
3214:
3211:
3186:
3182:
3159:
3155:
3143:
3140:
3139:
3135:
3134:
3117:
3114:
3111:
3108:
3105:
3102:
3096:
3093:
3090:
3084:
3081:
3075:
3072:
3069:
3063:
3060:
3054:
3051:
3048:
3042:
3039:
3033:
3030:
3027:
3021:
3018:
3012:
3009:
3006:
3000:
2993:
2992:
2991:
2974:
2971:
2968:
2964:
2960:
2957:
2951:
2948:
2945:
2939:
2936:
2930:
2927:
2924:
2918:
2915:
2909:
2906:
2903:
2897:
2894:
2888:
2885:
2882:
2876:
2869:
2868:
2867:
2814:
2811:
2808:
2790:
2787:
2786:
2785:
2776:with element
2760:
2757:
2754:
2729:
2725:
2704:
2701:
2695:
2692:
2686:
2683:
2680:
2669:
2665:
2644:
2641:
2635:
2632:
2626:
2623:
2620:
2609:
2605:
2590:
2588:
2585:
2583:
2580:
2578:
2575:
2572:
2571:
2568:
2565:
2562:
2560:
2557:
2555:
2552:
2549:
2548:
2545:
2542:
2540:
2537:
2534:
2532:
2529:
2526:
2525:
2522:
2519:
2517:
2514:
2512:
2509:
2506:
2503:
2502:
2498:
2495:
2492:
2489:
2487:
2486:
2483:
2467:
2463:
2451:
2448:
2447:
2443:
2441:
2425:
2421:
2393:
2388:
2380:
2377:
2374:
2363:
2352:
2349:
2346:
2338:
2334:
2330:
2324:
2321:
2318:
2310:
2306:
2294:
2286:
2283:
2277:
2274:
2271:
2260:
2256:
2248:
2243:
2235:
2232:
2229:
2218:
2207:
2204:
2201:
2193:
2189:
2185:
2179:
2176:
2173:
2165:
2161:
2149:
2141:
2138:
2132:
2129:
2126:
2115:
2111:
2103:
2098:
2090:
2087:
2084:
2073:
2062:
2059:
2056:
2048:
2044:
2040:
2034:
2031:
2028:
2020:
2016:
2004:
1996:
1993:
1987:
1984:
1981:
1970:
1966:
1954:
1953:
1952:
1935:
1932:
1929:
1918:
1900:
1896:
1865:
1861:
1838:
1834:
1822:
1819:
1818:
1817:
1815:
1814:
1797:
1794:
1791:
1787:
1783:
1780:
1774:
1771:
1768:
1762:
1759:
1753:
1750:
1747:
1741:
1721:
1693:
1689:
1682:
1679:
1676:
1668:
1664:
1660:
1654:
1651:
1648:
1642:
1639:
1633:
1630:
1627:
1621:
1598:
1595:
1594:
1593:
1569:
1565:
1544:
1541:
1535:
1532:
1529:
1521:
1517:
1502:
1499:
1496:
1493:
1490:
1487:
1486:
1482:
1479:
1476:
1473:
1470:
1467:
1466:
1462:
1459:
1456:
1453:
1450:
1447:
1446:
1442:
1439:
1436:
1433:
1430:
1427:
1426:
1422:
1419:
1416:
1413:
1410:
1407:
1406:
1402:
1399:
1396:
1393:
1390:
1388:
1387:
1384:
1382:
1366:
1362:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1314:
1300:
1297:
1296:
1292:
1290:
1276:
1268:
1267:
1250:
1242:
1240:
1223:
1215:
1213:
1212:Lactobacillus
1196:
1188:
1187:
1170:
1162:
1161:
1156:
1152:
1144:
1139:
1115:
1109:
1103:
1094:
1091:
1082:
1076:
1070:
1061:
1052:
1043:
1037:
1031:
1028:
1025:
1016:
993:
967:
964:
961:
935:
909:
886:
878:
858:
852:
846:
837:
834:
828:
822:
802:
779:
753:
730:
727:
724:
721:
718:
710:
689:
683:
677:
668:
662:
653:
647:
641:
632:
609:
603:
597:
586:
571:
568:
565:
545:
542:
536:
530:
522:
521:
520:
517:
497:
491:
485:
476:
453:
427:
416:
412:
396:
373:
367:
347:
344:
341:
338:
335:
332:
329:
326:
323:
300:
297:
294:
288:
268:
248:
245:
242:
234:
233:agglomerative
226:
224:
222:
218:
214:
210:
206:
202:
198:
179:
173:
170:
167:
161:
156:
153:
150:
147:
144:
141:
138:
130:
124:
121:
118:
112:
105:
104:
103:
101:
97:
93:
89:
85:
80:
78:
74:
68:
66:
57:
55:
53:
49:
43:
41:
37:
33:
19:
5308:(1): 54–64.
5305:
5301:
5295:
5276:
5270:
5251:
5247:
5234:
5217:
5213:
5207:
5172:
5166:
5131:
5121:
5112:
5106:
5081:
5075:
5056:
5050:
4857:
4515:
4422:
4411:
4383:
4257:
4134:
4124:
3897:
3894:
3787:
3604:
3531:
3521:
3511:
3505:((a,b),c,e)
3497:((a,b),c,e)
3461:
3146:
3141:
3131:
2989:
2793:
2788:
2596:
2586:
2581:
2576:
2566:
2558:
2553:
2543:
2538:
2530:
2520:
2515:
2510:
2454:
2449:
2412:
1916:
1825:
1820:
1811:
1601:
1596:
1508:
1383:
1303:
1298:
1264:
1239:Acholeplasma
1237:
1210:
1184:
1158:
1148:
518:
414:
413:is denoted (
410:
230:
220:
216:
212:
208:
204:
200:
196:
194:
99:
95:
91:
87:
83:
81:
72:
69:
64:
61:
44:
35:
29:
3489:matrix is:
2858:, and also
2444:Second step
1214:viridescens
469:is denoted
5042:References
4967:and space
4899:and space
3462:The final
3458:Final step
1293:First step
77:dendrogram
32:statistics
5173:Ribosomes
4949:
4745:λ
4685:π
4349:δ
4328:δ
4307:δ
4286:δ
4265:δ
4097:−
4073:δ
4070:−
4052:δ
4031:δ
4028:−
4010:δ
3989:δ
3986:−
3968:δ
3947:δ
3944:−
3926:δ
3905:δ
3846:δ
3795:δ
3237:and with
3109:−
3085:δ
3082:−
3064:δ
3043:δ
3040:−
3022:δ
3001:δ
2940:δ
2919:δ
2898:δ
2877:δ
1763:δ
1742:δ
1643:δ
1622:δ
345:−
336:…
246:×
154:∈
142:∈
48:astronomy
5357:Category
5000:See also
4255: :
3241: :
2482: :
5330:0242315
5322:2346439
5199:3241556
5158:2422630
5086:Bibcode
4435:), and
1263:), and
1241:modicum
5328:
5320:
5283:
5197:
5187:
5156:
5149:341310
5146:
5063:
4215:, and
2850:, and
2597:Here,
2504:(a,b)
2490:(a,b)
195:where
5318:JSTOR
5244:(PDF)
5036:WPGMA
5031:UPGMA
4504:UPGMA
4498:WPGMA
4433:WPGMA
4429:UPGMA
3233:with
2657:and
1884:with
5281:ISBN
5195:PMID
5185:ISBN
5154:PMID
5061:ISBN
4431:and
4416:for
4100:10.5
3745:and
3675:and
3605:Let
3582:and
3106:10.5
2975:10.5
2834:and
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666:min
135:min
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