Solving Non-Uniqueness in Agglomerative Hierarchical Clustering Using Multidendrograms

In agglomerative hierarchical clustering, pair-group methods suffer from a problem of non-uniqueness when two or more distances between different clusters coincide during the amalgamation process. The traditional approach for solving this drawback has been to take any arbitrary criterion in order to break ties between distances, which results in different hierarchical classifications depending on the criterion followed. In this article we propose a variable-group algorithm that consists in grouping more than two clusters at the same time when ties occur. We give a tree representation for the results of the algorithm, which we call a multidendrogram, as well as a generalization of the Lance andWilliams’ formula which enables the implementation of the algorithm in a recursive way. The authors thank A. Arenas for discussion and helpful comments. This work was partially supported by DGES of the Spanish Government Project No. FIS2006–13321–C02–02 and by a grant of Universitat Rovira i Virgili.     

Phenological studies of selected savanna mosses of south-western Nigeria

It was established that there is little diversity of bryophytes in the derived savanna. Mosses were found in the sampling sites, whereas liverworts were rarely observed. The reproductive methods of four dominant sexually reproducing savanna mosses —Archidium ohioense, Bryum coronatum, Fissidens minutifolius andTrachycarpidium tisserantii were monitored over two consecutive rainy seasons. Protonemal and gametophyte production were noticed in the field in March/April, and capsule dehiscence and spore dispersal occurred in September/October. The sequential stages of development, starting with gametangial production and ending with the falling of the dehisced capsules, occurred within the rainy season. However,A. ohioense andT. tisserantii did not discharge their spores easily (cleistocarpous), unlike the stegocarpous speciesB. coronatum andF. minutifolius. Water availability and possibly high humidity may have contributed to growth. The short period between sex organ formation and dehiscence of capsule seen in these studies, compared with the longer period in some temperate mosses, may be an advantage for bryophytes in a savanna environment.