4452:
2397:
4468:
4459:
4117:
4428:. 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.
4443:
1946:
2392:{\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}}}
4815:, 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
51:
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
59:
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
34:
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
4105:
4917:
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
3441:
4847:
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
3117:
4369:
2974:
4409:. 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.
31:. 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.
3889:
3880:
1693:
179:
1797:
2704:
2644:
347:
1544:
1124:
3732:
3662:
3569:
1341:
4954:
248:
4886:
4744:
4575:
4539:
3236:
695:
612:
864:
545:
303:
3476:
3220:
3188:
3161:
2817:
2763:
2731:
2469:
2427:
1938:
1902:
1867:
1840:
1571:
1368:
970:
730:
4983:
4915:
4842:
4813:
4684:
4604:
376:
996:
938:
912:
782:
756:
571:
456:
430:
2985:
4784:
4764:
4724:
4704:
4664:
4644:
4624:
4391:
4242:
4222:
4202:
4182:
4162:
4142:
3772:
3752:
3682:
3612:
3589:
1276:
1250:
1223:
1196:
1170:
886:
802:
503:
396:
268:
5069:
Feigelson, Eric (2012). "Classification in astronomy: past and present". In Way, Michael J.; Scargle, Jeffrey D.; Ali, Kamal M.; Srivastava, Ashok N. (eds.).
4249:
2861:
4100:{\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}
5177:
5351:
5273:
5053:
3779:
43:, which may often involve long strings of matter; in this application, it is also known as the friends-of-friends algorithm.
1606:
5268:. Philadelphia, Pa. Alexandria, Va: SIAM, Society for Industrial and Applied Mathematics American Statistical Association.
4116:
2429:
are not affected by the matrix update as they correspond to distances between elements not involved in the first cluster.
97:
1726:
4999:
4480:
4413:
1174:
940:
and adding a row and column corresponding to the newly formed cluster. The proximity between the new cluster, denoted
4844:. The SLINK algorithm then loops over the items, one by one, adding them to the representation of the clustering.
5004:
1139:
221:
28:
4402:
68:, which shows the sequence in which clusters were merged and the distance at which each merge took place.
4451:
4406:
224:
scheme that erases rows and columns in a proximity matrix as old clusters are merged into new ones. The
5104:. Developments in Environmental Modelling. Vol. 20 (Second English ed.). Amsterdam: Elsevier.
4505:
The naive algorithm for single-linkage clustering is easy to understand but slow, with time complexity
2649:
2589:
1951:
5074:
4626:
numbered items by two functions. These functions are both determined by finding the smallest cluster
308:
4786:. Storing these functions in two arrays that map each item number to its function value takes space
4849:
1501:
1001:
5306:
5160:
Olsen GJ (1988). "Phylogenetic analysis using ribosomal RNA". In Noller HF Jr, Moldave K (eds.).
3687:
3617:
3524:
3436:{\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}
1296:
1254:
4921:
227:
5269:
5183:
5173:
5142:
5049:
4855:
4729:
4544:
4508:
1148:
617:
579:
5298:
5244:
5229:
5210:
5165:
5132:
5124:
5082:
5014:
4994:
4425:
1143:
807:
515:
273:
5336:
5318:
3454:
3193:
3166:
3139:
3112:{\displaystyle \delta (u,v)=\delta (c,v)-\delta (a,u)=\delta (c,v)-\delta (b,u)=10.5-8.5=2}
2790:
2736:
2709:
2447:
2405:
1911:
1880:
1845:
1818:
1549:
1346:
943:
703:
5314:
5009:
4959:
4891:
4818:
4789:
4669:
4580:
4467:
4458:
352:
975:
917:
891:
761:
735:
550:
435:
409:
5289:
Gower JC, Ross GJ (1969). "Minimum spanning trees and single linkage cluster analysis".
5078:
4769:
4749:
4709:
4689:
4649:
4629:
4609:
4606:(both optimal) known as SLINK. The slink algorithm represents a clustering on a set of
4376:
4227:
4207:
4187:
4167:
4147:
4127:
3757:
3737:
3667:
3597:
3574:
1708:
1261:
1235:
1208:
1181:
1155:
871:
787:
461:
381:
253:
40:
5201:
Murtagh F, Contreras P (2012). "Algorithms for hierarchical clustering: an overview".
5169:
5137:
5116:
3190:(see below), reduced in size by two rows and two columns because of the clustering of
5345:
4432:
Comparison of dendrograms obtained under different clustering methods from the same
1869:(see below), reduced in size by one row and one column because of the clustering of
4364:{\displaystyle \delta (a,r)=\delta (b,r)=\delta (c,r)=\delta (e,r)=\delta (d,r)=14}
2444:
We now reiterate the three previous actions, starting from the new distance matrix
1227:
2827:
are now connected. Because of the ultrametricity constraint, the branches joining
4442:
5128:
65:
20:
5249:
4401:
The naive algorithm for single linkage clustering is essentially the same as
2969:{\displaystyle \delta (a,v)=\delta (b,v)=\delta (c,v)=\delta (e,v)=21/2=10.5}
5230:"SLINK: an optimally efficient algorithm for the single-link cluster method"
1200:
36:
576:
Find the most similar pair of clusters in the current clustering, say pair
52:
function used to determine the distance between two clusters, known as the
5187:
5146:
697:
where the minimum is over all pairs of clusters in the current clustering.
35:
classes that could usefully subdivide the data. However, it is popular in
5310:
5086:
5302:
5214:
5203:
4124:
The dendrogram is now complete. It is ultrametric because all tips (
5117:"Collection of published 5S, 5.8S and 4.5S ribosomal RNA sequences"
5024:
5019:
4492:
4486:
4421:
4417:
508:
The single linkage algorithm is composed of the following steps:
4541:. In 1973, R. Sibson proposed an algorithm with time complexity
56:, is what differentiates the agglomerative clustering methods.
1904:
correspond to the new distances, calculated by retaining the
888:, by deleting the rows and columns corresponding to clusters
4466:
4457:
4441:
4115:
1129:
If all objects are in one cluster, stop. Else, go to step 2.
5266:
Data clustering : theory, algorithms, and applications
4666:
and at least one larger-numbered item. The first function,
3875:{\displaystyle \delta (((a,b),c,e),r)=\delta (d,r)=28/2=14}
5164:. Methods in Enzymology. Vol. 164. pp. 793–812.
5071:
Advances in
Machine Learning and Data Mining for Astronomy
192:
are any two sets of elements considered as clusters, and
4766:
to the distance associated with the creation of cluster
1815:
We then proceed to update the initial proximity matrix
4852:, in a variation without binary heaps that takes time
1688:{\displaystyle \delta (a,u)=\delta (b,u)=D_{1}(a,b)/2}
64:. The result of the clustering can be visualized as a
4962:
4924:
4894:
4858:
4821:
4792:
4772:
4752:
4732:
4712:
4692:
4672:
4652:
4632:
4612:
4583:
4547:
4511:
4379:
4252:
4230:
4210:
4190:
4170:
4150:
4130:
3892:
3782:
3760:
3740:
3690:
3670:
3620:
3600:
3577:
3527:
3457:
3239:
3196:
3169:
3142:
2988:
2864:
2793:
2739:
2712:
2652:
2592:
2450:
2408:
1949:
1914:
1883:
1848:
1821:
1729:
1609:
1552:
1504:
1349:
1299:
1264:
1238:
1211:
1184:
1158:
1004:
978:
946:
920:
894:
874:
810:
790:
764:
738:
706:
620:
582:
553:
518:
464:
438:
412:
384:
355:
311:
276:
256:
230:
100:
71:
Mathematically, the linkage function – the distance
174:{\displaystyle D(X,Y)=\min _{x\in X,y\in Y}d(x,y),}
5291:Journal of the Royal Statistical Society, Series C
4977:
4948:
4909:
4880:
4836:
4807:
4778:
4758:
4738:
4718:
4698:
4678:
4658:
4638:
4618:
4598:
4569:
4533:
4385:
4363:
4236:
4216:
4196:
4176:
4156:
4136:
4099:
3874:
3766:
3746:
3726:
3676:
3656:
3606:
3583:
3563:
3470:
3435:
3214:
3182:
3155:
3111:
2968:
2811:
2757:
2725:
2698:
2638:
2463:
2421:
2391:
1932:
1896:
1861:
1834:
1792:{\displaystyle \delta (a,u)=\delta (b,u)=17/2=8.5}
1791:
1687:
1565:
1538:
1362:
1335:
1270:
1244:
1217:
1190:
1164:
1118:
990:
964:
932:
906:
880:
858:
796:
784:into a single cluster to form the next clustering
776:
750:
724:
689:
606:
565:
539:
497:
450:
424:
390:
370:
341:
297:
262:
242:
173:
512:Begin with the disjoint clustering having level
3403:
3298:
2357:
2288:
2212:
2143:
2067:
1998:
1044:
654:
305:. The clusterings are assigned sequence numbers
204:) denotes the distance between the two elements
123:
2855:are equal and have the following total length:
398:-th clustering. A cluster with sequence number
1707:. This corresponds to the expectation of the
8:
1113:
1047:
47:Overview of agglomerative clustering methods
16:Agglomerative hierarchical clustering method
1908:between each element of the first cluster
1142:genetic distance matrix computed from the
5248:
5136:
4961:
4923:
4893:
4869:
4857:
4820:
4791:
4771:
4751:
4731:
4711:
4691:
4671:
4651:
4631:
4611:
4582:
4558:
4546:
4522:
4510:
4378:
4251:
4229:
4209:
4189:
4169:
4149:
4129:
3891:
3858:
3781:
3759:
3739:
3689:
3669:
3619:
3599:
3576:
3526:
3462:
3456:
3376:
3348:
3308:
3244:
3238:
3195:
3174:
3168:
3147:
3141:
2987:
2952:
2863:
2792:
2738:
2717:
2711:
2657:
2651:
2597:
2591:
2455:
2449:
2413:
2407:
2326:
2298:
2248:
2181:
2153:
2103:
2036:
2008:
1958:
1950:
1948:
1913:
1888:
1882:
1853:
1847:
1826:
1820:
1775:
1728:
1677:
1656:
1608:
1557:
1551:
1509:
1503:
1354:
1348:
1298:
1293:Let us assume that we have five elements
1263:
1237:
1210:
1183:
1157:
1003:
977:
945:
919:
893:
873:
809:
789:
763:
737:
705:
619:
581:
552:
517:
463:
437:
411:
383:
354:
310:
275:
255:
229:
126:
99:
4706:to the largest-numbered item in cluster
4430:
3684:are now connected. The branches joining
3480:
2473:
1374:
5073:. Chapman and Hall/CRC. pp. 3–10.
5048:. Chichester, West Sussex, U.K: Wiley.
5036:
3884:We deduce the remaining branch length:
60:be merged. The method is also known as
5243:(1). British Computer Society: 30–34.
4373:The dendrogram is therefore rooted by
804:. Set the level of this clustering to
4433:
2979:We deduce the missing branch length:
1146:sequence alignment of five bacteria:
406:) and the proximity between clusters
7:
4412:Alternative linkage schemes include
1940:and each of the remaining elements:
1370:of pairwise distances between them:
1138:This working example is based on a
5209:(1). Wiley Online Library: 86–97.
3163:matrix into a new distance matrix
14:
4120:Single Linkage Dendrogram 5S data
2699:{\displaystyle D_{2}((a,b),e)=21}
2639:{\displaystyle D_{2}((a,b),c)=21}
1711:hypothesis. The branches joining
91:– is described by the expression
4450:
3614:denote the (root) node to which
3121:
1801:
5100:Legendre P, Legendre L (1998).
2778:Second branch length estimation
700:Increment the sequence number:
342:{\displaystyle 0,1,\ldots ,n-1}
5115:Erdmann VA, Wolters J (1986).
4972:
4966:
4943:
4928:
4904:
4898:
4875:
4862:
4831:
4825:
4802:
4796:
4593:
4587:
4564:
4551:
4528:
4515:
4416:, average linkage clustering (
4352:
4340:
4331:
4319:
4310:
4298:
4289:
4277:
4268:
4256:
4076:
4064:
4055:
4043:
4034:
4022:
4013:
4001:
3992:
3980:
3971:
3959:
3950:
3938:
3929:
3917:
3908:
3896:
3849:
3837:
3828:
3819:
3804:
3792:
3789:
3786:
3721:
3706:
3694:
3691:
3651:
3636:
3624:
3621:
3558:
3543:
3531:
3528:
3424:
3406:
3397:
3394:
3382:
3366:
3354:
3338:
3329:
3317:
3314:
3301:
3292:
3283:
3268:
3256:
3253:
3250:
3209:
3197:
3136:We then proceed to update the
3088:
3076:
3067:
3055:
3046:
3034:
3025:
3013:
3004:
2992:
2943:
2931:
2922:
2910:
2901:
2889:
2880:
2868:
2806:
2794:
2752:
2740:
2687:
2678:
2666:
2663:
2627:
2618:
2606:
2603:
2372:
2360:
2347:
2344:
2332:
2316:
2304:
2291:
2278:
2269:
2257:
2254:
2227:
2215:
2202:
2199:
2187:
2171:
2159:
2146:
2133:
2124:
2112:
2109:
2082:
2070:
2057:
2054:
2042:
2026:
2014:
2001:
1988:
1979:
1967:
1964:
1927:
1915:
1766:
1754:
1745:
1733:
1674:
1662:
1646:
1634:
1625:
1613:
1586:First branch length estimation
1527:
1515:
1330:
1300:
1110:
1107:
1101:
1095:
1089:
1086:
1077:
1074:
1068:
1062:
1056:
1053:
1038:
1035:
1029:
1023:
1011:
1008:
985:
979:
959:
947:
927:
921:
901:
895:
853:
850:
844:
838:
832:
829:
820:
814:
771:
765:
745:
739:
684:
681:
675:
669:
663:
660:
648:
645:
639:
633:
627:
624:
601:
595:
589:
583:
528:
522:
492:
489:
483:
477:
471:
468:
445:
439:
419:
413:
365:
359:
292:
280:
220:The following algorithm is an
165:
153:
116:
104:
1:
5170:10.1016/s0076-6879(88)64084-5
4646:that contains both item
4110:The single-linkage dendrogram
3131:Second distance matrix update
1539:{\displaystyle D_{1}(a,b)=17}
1119:{\displaystyle d=\min\{d,d\}}
868:Update the proximity matrix,
27:is one of several methods of
4491:Average linkage clustering:
4485:Average linkage clustering:
1842:into a new proximity matrix
1810:First distance matrix update
1603:are now connected. Setting
62:nearest neighbour clustering
5352:Cluster analysis algorithms
5123:. 14 Suppl (Suppl): r1-59.
5000:Complete-linkage clustering
4481:Complete-linkage clustering
4414:complete linkage clustering
3727:{\displaystyle ((a,b),c,e)}
3657:{\displaystyle ((a,b),c,e)}
3564:{\displaystyle ((a,b),c,e)}
1336:{\displaystyle (a,b,c,d,e)}
1175:Bacillus stearothermophilus
5368:
4949:{\displaystyle O(n\log n)}
4477:Single-linkage clustering
2787:denote the node to which
2706:are the lowest values of
1595:denote the node to which
1573:, so we cluster elements
1343:and the following matrix
243:{\displaystyle N\times N}
25:single-linkage clustering
4881:{\displaystyle O(n^{2})}
4739:{\displaystyle \lambda }
4570:{\displaystyle O(n^{2})}
4534:{\displaystyle O(n^{3})}
3122:see the final dendrogram
1802:see the final dendrogram
690:{\displaystyle d=\min d}
5337:Linkages used in Matlab
5129:10.1093/nar/14.suppl.r1
5005:Hierarchical clustering
4726:. The second function,
4224:) are equidistant from
1546:is the lowest value of
607:{\displaystyle (r),(s)}
270:contains all distances
29:hierarchical clustering
5250:10.1093/comjnl/16.1.30
5121:Nucleic Acids Research
4979:
4950:
4911:
4882:
4838:
4809:
4780:
4760:
4740:
4720:
4700:
4680:
4660:
4640:
4620:
4600:
4571:
4535:
4471:
4462:
4446:
4407:minimum spanning trees
4387:
4365:
4238:
4218:
4198:
4178:
4158:
4138:
4121:
4101:
3876:
3768:
3748:
3728:
3678:
3658:
3608:
3585:
3565:
3472:
3437:
3216:
3184:
3157:
3113:
2970:
2813:
2759:
2727:
2700:
2640:
2465:
2423:
2393:
1934:
1898:
1863:
1836:
1793:
1695:ensures that elements
1689:
1567:
1540:
1364:
1337:
1272:
1246:
1219:
1192:
1166:
1120:
992:
966:
934:
908:
882:
860:
859:{\displaystyle L(m)=d}
798:
778:
752:
726:
691:
608:
567:
541:
540:{\displaystyle L(0)=0}
499:
452:
426:
392:
372:
343:
299:
298:{\displaystyle d(i,j)}
264:
244:
175:
4980:
4951:
4912:
4883:
4839:
4810:
4781:
4761:
4741:
4721:
4701:
4681:
4661:
4641:
4621:
4601:
4577:and space complexity
4572:
4536:
4470:
4461:
4445:
4388:
4366:
4239:
4219:
4199:
4179:
4159:
4139:
4119:
4102:
3877:
3769:
3749:
3729:
3679:
3659:
3609:
3586:
3566:
3473:
3471:{\displaystyle D_{3}}
3438:
3217:
3215:{\displaystyle (a,b)}
3185:
3183:{\displaystyle D_{3}}
3158:
3156:{\displaystyle D_{2}}
3114:
2971:
2814:
2812:{\displaystyle (a,b)}
2760:
2758:{\displaystyle (a,b)}
2733:, so we join cluster
2728:
2726:{\displaystyle D_{2}}
2701:
2641:
2466:
2464:{\displaystyle D_{2}}
2424:
2422:{\displaystyle D_{2}}
2402:Italicized values in
2394:
1935:
1933:{\displaystyle (a,b)}
1899:
1897:{\displaystyle D_{2}}
1864:
1862:{\displaystyle D_{2}}
1837:
1835:{\displaystyle D_{1}}
1794:
1703:are equidistant from
1690:
1568:
1566:{\displaystyle D_{1}}
1541:
1365:
1363:{\displaystyle D_{1}}
1338:
1273:
1247:
1220:
1193:
1167:
1121:
993:
967:
965:{\displaystyle (r,s)}
935:
909:
883:
861:
799:
779:
753:
727:
725:{\displaystyle m=m+1}
692:
609:
568:
542:
500:
453:
427:
393:
373:
344:
300:
265:
245:
176:
5237:The Computer Journal
4978:{\displaystyle O(n)}
4960:
4922:
4910:{\displaystyle O(n)}
4892:
4856:
4837:{\displaystyle O(n)}
4819:
4808:{\displaystyle O(n)}
4790:
4770:
4750:
4730:
4710:
4690:
4679:{\displaystyle \pi }
4670:
4650:
4630:
4610:
4599:{\displaystyle O(n)}
4581:
4545:
4509:
4393:, its deepest node.
4377:
4250:
4228:
4208:
4188:
4168:
4148:
4128:
3890:
3780:
3758:
3738:
3688:
3668:
3618:
3598:
3575:
3525:
3521:So we join clusters
3455:
3237:
3194:
3167:
3140:
2986:
2862:
2791:
2737:
2710:
2650:
2590:
2448:
2406:
1947:
1912:
1881:
1846:
1819:
1727:
1607:
1550:
1502:
1347:
1297:
1262:
1236:
1209:
1182:
1156:
1002:
976:
944:
918:
892:
872:
808:
788:
762:
736:
704:
618:
580:
551:
547:and sequence number
516:
462:
436:
410:
382:
378:is the level of the
371:{\displaystyle L(k)}
353:
309:
274:
254:
228:
98:
5079:2012amld.book....3F
4437:
4403:Kruskal's algorithm
3774:then have lengths:
1723:then have lengths
991:{\displaystyle (k)}
972:and an old cluster
933:{\displaystyle (s)}
907:{\displaystyle (r)}
777:{\displaystyle (s)}
751:{\displaystyle (r)}
566:{\displaystyle m=0}
451:{\displaystyle (s)}
425:{\displaystyle (r)}
83:) between clusters
5044:Everitt B (2011).
4975:
4946:
4907:
4878:
4834:
4805:
4776:
4756:
4736:
4716:
4696:
4676:
4656:
4636:
4616:
4596:
4567:
4531:
4472:
4463:
4447:
4431:
4383:
4361:
4234:
4214:
4194:
4174:
4154:
4134:
4122:
4097:
3872:
3764:
3744:
3724:
3674:
3654:
3604:
3581:
3561:
3468:
3433:
3212:
3180:
3153:
3109:
2966:
2809:
2755:
2723:
2696:
2636:
2461:
2419:
2389:
2387:
1930:
1894:
1859:
1832:
1789:
1685:
1563:
1536:
1360:
1333:
1268:
1255:Micrococcus luteus
1242:
1215:
1188:
1162:
1116:
988:
962:
930:
904:
878:
856:
794:
774:
748:
722:
687:
604:
563:
537:
495:
448:
422:
388:
368:
339:
295:
260:
240:
171:
149:
5228:Sibson R (1973).
5179:978-0-12-182065-7
5102:Numerical Ecology
4779:{\displaystyle C}
4759:{\displaystyle i}
4746:, maps item
4719:{\displaystyle C}
4699:{\displaystyle i}
4686:, maps item
4659:{\displaystyle i}
4639:{\displaystyle C}
4619:{\displaystyle n}
4501:Faster algorithms
4498:
4497:
4386:{\displaystyle r}
4237:{\displaystyle r}
4217:{\displaystyle d}
4197:{\displaystyle e}
4177:{\displaystyle c}
4157:{\displaystyle b}
4137:{\displaystyle a}
3767:{\displaystyle r}
3747:{\displaystyle d}
3677:{\displaystyle d}
3607:{\displaystyle r}
3584:{\displaystyle d}
3519:
3518:
2769:and with element
2584:
2583:
2439:Second clustering
1877:. Bold values in
1498:In this example,
1496:
1495:
1271:{\displaystyle e}
1245:{\displaystyle d}
1218:{\displaystyle c}
1191:{\displaystyle b}
1165:{\displaystyle a}
1149:Bacillus subtilis
881:{\displaystyle D}
797:{\displaystyle m}
732:. Merge clusters
498:{\displaystyle d}
391:{\displaystyle k}
263:{\displaystyle D}
250:proximity matrix
122:
5359:
5324:
5322:
5286:
5280:
5279:
5261:
5255:
5254:
5252:
5234:
5225:
5219:
5218:
5198:
5192:
5191:
5157:
5151:
5150:
5140:
5112:
5106:
5105:
5097:
5091:
5090:
5087:10.1201/b11822-7
5066:
5060:
5059:
5046:Cluster analysis
5041:
5015:Neighbor-joining
4995:Cluster analysis
4984:
4982:
4981:
4976:
4955:
4953:
4952:
4947:
4916:
4914:
4913:
4908:
4887:
4885:
4884:
4879:
4874:
4873:
4850:Prim's algorithm
4843:
4841:
4840:
4835:
4814:
4812:
4811:
4806:
4785:
4783:
4782:
4777:
4765:
4763:
4762:
4757:
4745:
4743:
4742:
4737:
4725:
4723:
4722:
4717:
4705:
4703:
4702:
4697:
4685:
4683:
4682:
4677:
4665:
4663:
4662:
4657:
4645:
4643:
4642:
4637:
4625:
4623:
4622:
4617:
4605:
4603:
4602:
4597:
4576:
4574:
4573:
4568:
4563:
4562:
4540:
4538:
4537:
4532:
4527:
4526:
4454:
4438:
4392:
4390:
4389:
4384:
4370:
4368:
4367:
4362:
4243:
4241:
4240:
4235:
4223:
4221:
4220:
4215:
4203:
4201:
4200:
4195:
4183:
4181:
4180:
4175:
4163:
4161:
4160:
4155:
4143:
4141:
4140:
4135:
4106:
4104:
4103:
4098:
3881:
3879:
3878:
3873:
3862:
3773:
3771:
3770:
3765:
3753:
3751:
3750:
3745:
3733:
3731:
3730:
3725:
3683:
3681:
3680:
3675:
3663:
3661:
3660:
3655:
3613:
3611:
3610:
3605:
3590:
3588:
3587:
3582:
3570:
3568:
3567:
3562:
3481:
3477:
3475:
3474:
3469:
3467:
3466:
3442:
3440:
3439:
3434:
3381:
3380:
3353:
3352:
3313:
3312:
3249:
3248:
3229:
3225:
3221:
3219:
3218:
3213:
3189:
3187:
3186:
3181:
3179:
3178:
3162:
3160:
3159:
3154:
3152:
3151:
3118:
3116:
3115:
3110:
2975:
2973:
2972:
2967:
2956:
2854:
2850:
2846:
2842:
2838:
2834:
2830:
2826:
2822:
2818:
2816:
2815:
2810:
2786:
2772:
2768:
2764:
2762:
2761:
2756:
2732:
2730:
2729:
2724:
2722:
2721:
2705:
2703:
2702:
2697:
2662:
2661:
2645:
2643:
2642:
2637:
2602:
2601:
2474:
2470:
2468:
2467:
2462:
2460:
2459:
2428:
2426:
2425:
2420:
2418:
2417:
2398:
2396:
2395:
2390:
2388:
2331:
2330:
2303:
2302:
2253:
2252:
2186:
2185:
2158:
2157:
2108:
2107:
2041:
2040:
2013:
2012:
1963:
1962:
1939:
1937:
1936:
1931:
1906:minimum distance
1903:
1901:
1900:
1895:
1893:
1892:
1876:
1872:
1868:
1866:
1865:
1860:
1858:
1857:
1841:
1839:
1838:
1833:
1831:
1830:
1798:
1796:
1795:
1790:
1779:
1722:
1718:
1714:
1706:
1702:
1698:
1694:
1692:
1691:
1686:
1681:
1661:
1660:
1602:
1598:
1594:
1580:
1576:
1572:
1570:
1569:
1564:
1562:
1561:
1545:
1543:
1542:
1537:
1514:
1513:
1375:
1369:
1367:
1366:
1361:
1359:
1358:
1342:
1340:
1339:
1334:
1288:First clustering
1277:
1275:
1274:
1269:
1251:
1249:
1248:
1243:
1224:
1222:
1221:
1216:
1197:
1195:
1194:
1189:
1171:
1169:
1168:
1163:
1144:5S ribosomal RNA
1125:
1123:
1122:
1117:
997:
995:
994:
989:
971:
969:
968:
963:
939:
937:
936:
931:
913:
911:
910:
905:
887:
885:
884:
879:
865:
863:
862:
857:
803:
801:
800:
795:
783:
781:
780:
775:
757:
755:
754:
749:
731:
729:
728:
723:
696:
694:
693:
688:
613:
611:
610:
605:
572:
570:
569:
564:
546:
544:
543:
538:
504:
502:
501:
496:
457:
455:
454:
449:
431:
429:
428:
423:
397:
395:
394:
389:
377:
375:
374:
369:
348:
346:
345:
340:
304:
302:
301:
296:
269:
267:
266:
261:
249:
247:
246:
241:
180:
178:
177:
172:
148:
54:linkage function
5367:
5366:
5362:
5361:
5360:
5358:
5357:
5356:
5342:
5341:
5333:
5328:
5327:
5303:10.2307/2346439
5288:
5287:
5283:
5276:
5263:
5262:
5258:
5232:
5227:
5226:
5222:
5215:10.1002/widm.53
5200:
5199:
5195:
5180:
5159:
5158:
5154:
5114:
5113:
5109:
5099:
5098:
5094:
5068:
5067:
5063:
5056:
5043:
5042:
5038:
5033:
5010:Molecular clock
4991:
4958:
4957:
4920:
4919:
4890:
4889:
4865:
4854:
4853:
4817:
4816:
4788:
4787:
4768:
4767:
4748:
4747:
4728:
4727:
4708:
4707:
4688:
4687:
4668:
4667:
4648:
4647:
4628:
4627:
4608:
4607:
4579:
4578:
4554:
4543:
4542:
4518:
4507:
4506:
4503:
4434:distance matrix
4399:
4375:
4374:
4248:
4247:
4226:
4225:
4206:
4205:
4186:
4185:
4166:
4165:
4146:
4145:
4126:
4125:
4112:
3888:
3887:
3778:
3777:
3756:
3755:
3736:
3735:
3686:
3685:
3666:
3665:
3616:
3615:
3596:
3595:
3573:
3572:
3523:
3522:
3458:
3453:
3452:
3449:
3372:
3344:
3304:
3240:
3235:
3234:
3227:
3223:
3192:
3191:
3170:
3165:
3164:
3143:
3138:
3137:
2984:
2983:
2860:
2859:
2852:
2848:
2844:
2840:
2836:
2832:
2828:
2824:
2820:
2789:
2788:
2784:
2770:
2766:
2735:
2734:
2713:
2708:
2707:
2653:
2648:
2647:
2593:
2588:
2587:
2451:
2446:
2445:
2435:
2409:
2404:
2403:
2386:
2385:
2380:
2375:
2355:
2350:
2322:
2294:
2286:
2281:
2244:
2241:
2240:
2235:
2230:
2210:
2205:
2177:
2149:
2141:
2136:
2099:
2096:
2095:
2090:
2085:
2065:
2060:
2032:
2004:
1996:
1991:
1954:
1945:
1944:
1910:
1909:
1884:
1879:
1878:
1874:
1870:
1849:
1844:
1843:
1822:
1817:
1816:
1725:
1724:
1720:
1716:
1712:
1704:
1700:
1696:
1652:
1605:
1604:
1600:
1596:
1592:
1578:
1574:
1553:
1548:
1547:
1505:
1500:
1499:
1350:
1345:
1344:
1295:
1294:
1284:
1260:
1259:
1234:
1233:
1207:
1206:
1180:
1179:
1154:
1153:
1136:
1134:Working example
1000:
999:
974:
973:
942:
941:
916:
915:
890:
889:
870:
869:
806:
805:
786:
785:
760:
759:
734:
733:
702:
701:
616:
615:
614:, according to
578:
577:
549:
548:
514:
513:
460:
459:
434:
433:
408:
407:
380:
379:
351:
350:
307:
306:
272:
271:
252:
251:
226:
225:
218:
216:Naive algorithm
96:
95:
49:
41:galaxy clusters
17:
12:
11:
5:
5365:
5363:
5355:
5354:
5344:
5343:
5340:
5339:
5332:
5331:External links
5329:
5326:
5325:
5281:
5274:
5264:Gan G (2007).
5256:
5220:
5193:
5178:
5152:
5107:
5092:
5061:
5054:
5035:
5034:
5032:
5029:
5028:
5027:
5022:
5017:
5012:
5007:
5002:
4997:
4990:
4987:
4974:
4971:
4968:
4965:
4945:
4942:
4939:
4936:
4933:
4930:
4927:
4906:
4903:
4900:
4897:
4877:
4872:
4868:
4864:
4861:
4833:
4830:
4827:
4824:
4804:
4801:
4798:
4795:
4775:
4755:
4735:
4715:
4695:
4675:
4655:
4635:
4615:
4595:
4592:
4589:
4586:
4566:
4561:
4557:
4553:
4550:
4530:
4525:
4521:
4517:
4514:
4502:
4499:
4496:
4495:
4489:
4483:
4478:
4474:
4473:
4464:
4455:
4448:
4398:
4397:Other linkages
4395:
4382:
4360:
4357:
4354:
4351:
4348:
4345:
4342:
4339:
4336:
4333:
4330:
4327:
4324:
4321:
4318:
4315:
4312:
4309:
4306:
4303:
4300:
4297:
4294:
4291:
4288:
4285:
4282:
4279:
4276:
4273:
4270:
4267:
4264:
4261:
4258:
4255:
4233:
4213:
4193:
4173:
4153:
4133:
4111:
4108:
4096:
4093:
4090:
4087:
4084:
4081:
4078:
4075:
4072:
4069:
4066:
4063:
4060:
4057:
4054:
4051:
4048:
4045:
4042:
4039:
4036:
4033:
4030:
4027:
4024:
4021:
4018:
4015:
4012:
4009:
4006:
4003:
4000:
3997:
3994:
3991:
3988:
3985:
3982:
3979:
3976:
3973:
3970:
3967:
3964:
3961:
3958:
3955:
3952:
3949:
3946:
3943:
3940:
3937:
3934:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3910:
3907:
3904:
3901:
3898:
3895:
3871:
3868:
3865:
3861:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3833:
3830:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3806:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3763:
3743:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3673:
3653:
3650:
3647:
3644:
3641:
3638:
3635:
3632:
3629:
3626:
3623:
3603:
3580:
3560:
3557:
3554:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3530:
3517:
3516:
3513:
3508:
3504:
3503:
3498:
3495:
3491:
3490:
3487:
3484:
3465:
3461:
3448:
3445:
3444:
3443:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3379:
3375:
3371:
3368:
3365:
3362:
3359:
3356:
3351:
3347:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3322:
3319:
3316:
3311:
3307:
3303:
3300:
3297:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3247:
3243:
3211:
3208:
3205:
3202:
3199:
3177:
3173:
3150:
3146:
3134:
3133:
3127:
3126:
3108:
3105:
3102:
3099:
3096:
3093:
3090:
3087:
3084:
3081:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3015:
3012:
3009:
3006:
3003:
3000:
2997:
2994:
2991:
2977:
2976:
2965:
2962:
2959:
2955:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2808:
2805:
2802:
2799:
2796:
2781:
2780:
2754:
2751:
2748:
2745:
2742:
2720:
2716:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2660:
2656:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2600:
2596:
2582:
2581:
2578:
2573:
2568:
2563:
2559:
2558:
2553:
2550:
2545:
2540:
2536:
2535:
2530:
2525:
2522:
2517:
2513:
2512:
2507:
2502:
2497:
2494:
2490:
2489:
2486:
2483:
2480:
2477:
2458:
2454:
2442:
2441:
2434:
2431:
2416:
2412:
2400:
2399:
2384:
2381:
2379:
2376:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2354:
2351:
2349:
2346:
2343:
2340:
2337:
2334:
2329:
2325:
2321:
2318:
2315:
2312:
2309:
2306:
2301:
2297:
2293:
2290:
2287:
2285:
2282:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2251:
2247:
2243:
2242:
2239:
2236:
2234:
2231:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2209:
2206:
2204:
2201:
2198:
2195:
2192:
2189:
2184:
2180:
2176:
2173:
2170:
2167:
2164:
2161:
2156:
2152:
2148:
2145:
2142:
2140:
2137:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2106:
2102:
2098:
2097:
2094:
2091:
2089:
2086:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2064:
2061:
2059:
2056:
2053:
2050:
2047:
2044:
2039:
2035:
2031:
2028:
2025:
2022:
2019:
2016:
2011:
2007:
2003:
2000:
1997:
1995:
1992:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1961:
1957:
1953:
1952:
1929:
1926:
1923:
1920:
1917:
1891:
1887:
1856:
1852:
1829:
1825:
1813:
1812:
1788:
1785:
1782:
1778:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1709:ultrametricity
1684:
1680:
1676:
1673:
1670:
1667:
1664:
1659:
1655:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1589:
1588:
1560:
1556:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1512:
1508:
1494:
1493:
1490:
1487:
1484:
1481:
1478:
1474:
1473:
1470:
1467:
1464:
1461:
1458:
1454:
1453:
1450:
1447:
1444:
1441:
1438:
1434:
1433:
1430:
1427:
1424:
1421:
1418:
1414:
1413:
1410:
1407:
1404:
1401:
1398:
1394:
1393:
1390:
1387:
1384:
1381:
1378:
1357:
1353:
1332:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1302:
1291:
1290:
1283:
1280:
1267:
1241:
1214:
1187:
1161:
1135:
1132:
1131:
1130:
1127:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
998:is defined as
987:
984:
981:
961:
958:
955:
952:
949:
929:
926:
923:
903:
900:
897:
877:
866:
855:
852:
849:
846:
843:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
793:
773:
770:
767:
747:
744:
741:
721:
718:
715:
712:
709:
698:
686:
683:
680:
677:
674:
671:
668:
665:
662:
659:
656:
653:
650:
647:
644:
641:
638:
635:
632:
629:
626:
623:
603:
600:
597:
594:
591:
588:
585:
574:
562:
559:
556:
536:
533:
530:
527:
524:
521:
494:
491:
488:
485:
482:
479:
476:
473:
470:
467:
447:
444:
441:
421:
418:
415:
387:
367:
364:
361:
358:
338:
335:
332:
329:
326:
323:
320:
317:
314:
294:
291:
288:
285:
282:
279:
259:
239:
236:
233:
217:
214:
182:
181:
170:
167:
164:
161:
158:
155:
152:
147:
144:
141:
138:
135:
132:
129:
125:
121:
118:
115:
112:
109:
106:
103:
48:
45:
39:for analyzing
15:
13:
10:
9:
6:
4:
3:
2:
5364:
5353:
5350:
5349:
5347:
5338:
5335:
5334:
5330:
5320:
5316:
5312:
5308:
5304:
5300:
5296:
5292:
5285:
5282:
5277:
5275:9780898716238
5271:
5267:
5260:
5257:
5251:
5246:
5242:
5238:
5231:
5224:
5221:
5216:
5212:
5208:
5204:
5197:
5194:
5189:
5185:
5181:
5175:
5171:
5167:
5163:
5156:
5153:
5148:
5144:
5139:
5134:
5130:
5126:
5122:
5118:
5111:
5108:
5103:
5096:
5093:
5088:
5084:
5080:
5076:
5072:
5065:
5062:
5057:
5055:9780470749913
5051:
5047:
5040:
5037:
5030:
5026:
5023:
5021:
5018:
5016:
5013:
5011:
5008:
5006:
5003:
5001:
4998:
4996:
4993:
4992:
4988:
4986:
4969:
4963:
4940:
4937:
4934:
4931:
4925:
4901:
4895:
4870:
4866:
4859:
4851:
4845:
4828:
4822:
4799:
4793:
4773:
4753:
4733:
4713:
4693:
4673:
4653:
4633:
4613:
4590:
4584:
4559:
4555:
4548:
4523:
4519:
4512:
4500:
4494:
4490:
4488:
4484:
4482:
4479:
4476:
4475:
4469:
4465:
4460:
4456:
4453:
4449:
4444:
4440:
4439:
4435:
4429:
4427:
4426:Ward's method
4423:
4419:
4415:
4410:
4408:
4404:
4396:
4394:
4380:
4371:
4358:
4355:
4349:
4346:
4343:
4337:
4334:
4328:
4325:
4322:
4316:
4313:
4307:
4304:
4301:
4295:
4292:
4286:
4283:
4280:
4274:
4271:
4265:
4262:
4259:
4253:
4245:
4231:
4211:
4191:
4171:
4151:
4131:
4118:
4114:
4109:
4107:
4094:
4091:
4088:
4085:
4082:
4079:
4073:
4070:
4067:
4061:
4058:
4052:
4049:
4046:
4040:
4037:
4031:
4028:
4025:
4019:
4016:
4010:
4007:
4004:
3998:
3995:
3989:
3986:
3983:
3977:
3974:
3968:
3965:
3962:
3956:
3953:
3947:
3944:
3941:
3935:
3932:
3926:
3923:
3920:
3914:
3911:
3905:
3902:
3899:
3893:
3885:
3882:
3869:
3866:
3863:
3859:
3855:
3852:
3846:
3843:
3840:
3834:
3831:
3825:
3822:
3816:
3813:
3810:
3807:
3801:
3798:
3795:
3783:
3775:
3761:
3741:
3718:
3715:
3712:
3709:
3703:
3700:
3697:
3671:
3648:
3645:
3642:
3639:
3633:
3630:
3627:
3601:
3592:
3578:
3555:
3552:
3549:
3546:
3540:
3537:
3534:
3514:
3512:
3509:
3506:
3505:
3502:
3499:
3496:
3493:
3492:
3488:
3485:
3483:
3482:
3479:
3463:
3459:
3446:
3430:
3427:
3421:
3418:
3415:
3412:
3409:
3400:
3391:
3388:
3385:
3377:
3373:
3369:
3363:
3360:
3357:
3349:
3345:
3341:
3335:
3332:
3326:
3323:
3320:
3309:
3305:
3295:
3289:
3286:
3280:
3277:
3274:
3271:
3265:
3262:
3259:
3245:
3241:
3233:
3232:
3231:
3206:
3203:
3200:
3175:
3171:
3148:
3144:
3132:
3129:
3128:
3124:
3123:
3106:
3103:
3100:
3097:
3094:
3091:
3085:
3082:
3079:
3073:
3070:
3064:
3061:
3058:
3052:
3049:
3043:
3040:
3037:
3031:
3028:
3022:
3019:
3016:
3010:
3007:
3001:
2998:
2995:
2989:
2982:
2981:
2980:
2963:
2960:
2957:
2953:
2949:
2946:
2940:
2937:
2934:
2928:
2925:
2919:
2916:
2913:
2907:
2904:
2898:
2895:
2892:
2886:
2883:
2877:
2874:
2871:
2865:
2858:
2857:
2856:
2803:
2800:
2797:
2779:
2776:
2775:
2774:
2765:with element
2749:
2746:
2743:
2718:
2714:
2693:
2690:
2684:
2681:
2675:
2672:
2669:
2658:
2654:
2633:
2630:
2624:
2621:
2615:
2612:
2609:
2598:
2594:
2579:
2577:
2574:
2572:
2569:
2567:
2564:
2561:
2560:
2557:
2554:
2551:
2549:
2546:
2544:
2541:
2538:
2537:
2534:
2531:
2529:
2526:
2523:
2521:
2518:
2515:
2514:
2511:
2508:
2506:
2503:
2501:
2498:
2495:
2492:
2491:
2487:
2484:
2481:
2478:
2476:
2475:
2472:
2456:
2452:
2440:
2437:
2436:
2432:
2430:
2414:
2410:
2382:
2377:
2369:
2366:
2363:
2352:
2341:
2338:
2335:
2327:
2323:
2319:
2313:
2310:
2307:
2299:
2295:
2283:
2275:
2272:
2266:
2263:
2260:
2249:
2245:
2237:
2232:
2224:
2221:
2218:
2207:
2196:
2193:
2190:
2182:
2178:
2174:
2168:
2165:
2162:
2154:
2150:
2138:
2130:
2127:
2121:
2118:
2115:
2104:
2100:
2092:
2087:
2079:
2076:
2073:
2062:
2051:
2048:
2045:
2037:
2033:
2029:
2023:
2020:
2017:
2009:
2005:
1993:
1985:
1982:
1976:
1973:
1970:
1959:
1955:
1943:
1942:
1941:
1924:
1921:
1918:
1907:
1889:
1885:
1854:
1850:
1827:
1823:
1811:
1808:
1807:
1806:
1804:
1803:
1786:
1783:
1780:
1776:
1772:
1769:
1763:
1760:
1757:
1751:
1748:
1742:
1739:
1736:
1730:
1710:
1682:
1678:
1671:
1668:
1665:
1657:
1653:
1649:
1643:
1640:
1637:
1631:
1628:
1622:
1619:
1616:
1610:
1587:
1584:
1583:
1582:
1558:
1554:
1533:
1530:
1524:
1521:
1518:
1510:
1506:
1491:
1488:
1485:
1482:
1479:
1476:
1475:
1471:
1468:
1465:
1462:
1459:
1456:
1455:
1451:
1448:
1445:
1442:
1439:
1436:
1435:
1431:
1428:
1425:
1422:
1419:
1416:
1415:
1411:
1408:
1405:
1402:
1399:
1396:
1395:
1391:
1388:
1385:
1382:
1379:
1377:
1376:
1373:
1371:
1355:
1351:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1289:
1286:
1285:
1281:
1279:
1265:
1257:
1256:
1239:
1231:
1229:
1212:
1204:
1202:
1201:Lactobacillus
1185:
1177:
1176:
1159:
1151:
1150:
1145:
1141:
1133:
1128:
1104:
1098:
1092:
1083:
1080:
1071:
1065:
1059:
1050:
1041:
1032:
1026:
1020:
1017:
1014:
1005:
982:
956:
953:
950:
924:
898:
875:
867:
847:
841:
835:
826:
823:
817:
811:
791:
768:
742:
719:
716:
713:
710:
707:
699:
678:
672:
666:
657:
651:
642:
636:
630:
621:
598:
592:
586:
575:
560:
557:
554:
534:
531:
525:
519:
511:
510:
509:
506:
486:
480:
474:
465:
442:
416:
405:
401:
385:
362:
356:
336:
333:
330:
327:
324:
321:
318:
315:
312:
289:
286:
283:
277:
257:
237:
234:
231:
223:
222:agglomerative
215:
213:
211:
207:
203:
199:
195:
191:
187:
168:
162:
159:
156:
150:
145:
142:
139:
136:
133:
130:
127:
119:
113:
110:
107:
101:
94:
93:
92:
90:
86:
82:
78:
74:
69:
67:
63:
57:
55:
46:
44:
42:
38:
32:
30:
26:
22:
5297:(1): 54–64.
5294:
5290:
5284:
5265:
5259:
5240:
5236:
5223:
5206:
5202:
5196:
5161:
5155:
5120:
5110:
5101:
5095:
5070:
5064:
5045:
5039:
4846:
4504:
4411:
4400:
4372:
4246:
4123:
4113:
3886:
3883:
3776:
3593:
3520:
3510:
3500:
3494:((a,b),c,e)
3486:((a,b),c,e)
3450:
3135:
3130:
3120:
2978:
2782:
2777:
2585:
2575:
2570:
2565:
2555:
2547:
2542:
2532:
2527:
2519:
2509:
2504:
2499:
2443:
2438:
2401:
1905:
1814:
1809:
1800:
1590:
1585:
1497:
1372:
1292:
1287:
1253:
1228:Acholeplasma
1226:
1199:
1173:
1147:
1137:
507:
403:
402:is denoted (
399:
219:
209:
205:
201:
197:
193:
189:
185:
183:
88:
84:
80:
76:
72:
70:
61:
58:
53:
50:
33:
24:
18:
3478:matrix is:
2847:, and also
2433:Second step
1203:viridescens
458:is denoted
5031:References
4956:and space
4888:and space
3451:The final
3447:Final step
1282:First step
66:dendrogram
21:statistics
5162:Ribosomes
4938:
4734:λ
4674:π
4338:δ
4317:δ
4296:δ
4275:δ
4254:δ
4086:−
4062:δ
4059:−
4041:δ
4020:δ
4017:−
3999:δ
3978:δ
3975:−
3957:δ
3936:δ
3933:−
3915:δ
3894:δ
3835:δ
3784:δ
3226:and with
3098:−
3074:δ
3071:−
3053:δ
3032:δ
3029:−
3011:δ
2990:δ
2929:δ
2908:δ
2887:δ
2866:δ
1752:δ
1731:δ
1632:δ
1611:δ
334:−
325:…
235:×
143:∈
131:∈
37:astronomy
5346:Category
4989:See also
4244: :
3230: :
2471: :
5319:0242315
5311:2346439
5188:3241556
5147:2422630
5075:Bibcode
4424:), and
1252:), and
1230:modicum
5317:
5309:
5272:
5186:
5176:
5145:
5138:341310
5135:
5052:
4204:, and
2839:, and
2586:Here,
2493:(a,b)
2479:(a,b)
184:where
5307:JSTOR
5233:(PDF)
5025:WPGMA
5020:UPGMA
4493:UPGMA
4487:WPGMA
4422:WPGMA
4418:UPGMA
3222:with
2646:and
1873:with
5270:ISBN
5184:PMID
5174:ISBN
5143:PMID
5050:ISBN
4420:and
4405:for
4089:10.5
3734:and
3664:and
3594:Let
3571:and
3095:10.5
2964:10.5
2823:and
2783:Let
1715:and
1699:and
1599:and
1591:Let
1577:and
1140:JC69
914:and
758:and
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5245:doi
5211:doi
5166:doi
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3404:min
3299:min
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