JOURNAL OF AGRICULTURAL SCIENCE IN FINLAND Maataloustieteellinen Aikakauskirja Vol. 60: 327—446, 1988 Incompatibility classes and fruit set in natural populations of arctic bramble (Rubus arcticus L.) in Finland JUSSI TAMMISOLA Department of Plant Breeding, University of Helsinki SF-00710 Helsinki, Finland Abstract. In Finnish natural populations, arctic bramble proved uniformly self-incom- patible. In vigorous and richly flowering populations, the intensity of fruit set is governed by the number of incompatibility classes present. Most non-fruiting populations contain only one incompatibility class (and most likely only one clone) and therefore totally lack compatible pollen. Richly fruiting populations usually contain at least three incompatibility classes. A clone with an estimated size of 80 metres and age of 160 years was found. This supports the vegetativeburst explanation for the “sudden appearance” of arctic bramble populations. For the purposes of plant breeding, a large genetic collection is required. In cultivation, a thor- ough mixture of at least three varieties is recommended. The study was dynamically optimized. A computerized guidance system was constructed, which analyzed the accumulating results and yieldedrecommendations for forthcoming cross- es. For the analysis into equivalence classes of incompatibility, a stepwise clustering algorithm of the single move type, based on the maximum likelihood principle, was introduced. This partitioning was based on the number of seeds in a cross, considering it as a stochastic varia- ble. Seed number probability distributions in intraclass and interclass pollinations were acquired utilizing non-parametric density estimation. Finally, both incompatibility class and seed prob- ability estimates were adjusted together iteratively. A recommendation algorithm was produced, based on a partially heuristic principle of maximum lability maximum stability. With the aid of the guidance system, incompatibility classes could be resolved using about one tenth of the number of crosses required in a conventional system. Index words: Arctic bramble, Rubus arclicus, incompatibility, fruit set, wild berries, population structure, clonal pattern, plant breeding, reproductiveecology, dynamic programming, stepwise clustering, density estimation, itera- tion, artificial intelligence 327 https://www.c-info.fi/en/info/?token=gKPHbNUeuE-bQaUm.Aof_7oFAE-zU3o_07vGoWw.RsZ14IOQqYTfbEf7ydBQlZ58HnrNkdXmrslduDu6L5rwSkbbQcbZpDSYCOIUntN6UAHal7mN-9bgY8zo0Pe9Ne7rMwhGhzjp-c1uwjPgf8X4HBjXG8Qn2oQmpY_8p4rfV1pcA038RfOraABbyeROLJ88Uq3jt8qLpt6wRfo 329 Contents ABSTRACT 327 1. INTRODUCTION 333 1.1. Characterization of Rubus arcticus 333 1.1.1. Taxonomy and distribution 333 1.1.2. Aroma 333 1.1.3. Habitats 335 1.1.4. Spread through growth and reproduction 337 1.1.5. Pollination 339 1.1.6. Self-sterility 341 1.2. Variation of fruit set in nature 342 1.3. Hypotheses and study plan 344 2. MATERIAL AND METHODS 345 2.1. Sampling of populations 345 2.2. Sampling of shoots 345 2.2.1. Number of ramets to be taken in a population 345 2.2.2. Randomization 349 2.3. Observations in natural populations 350 2.4. Management of the sampled ramets 350 2.5. Greenhouse in 1976—78 351 2.6. Experimental field in 1977—80 351 2.7. Techniques in crossing 352 2.8. Cytological staining method 353 2.9. Statistical test methods 353 3. RECURSIVE SYSTEM OF EXPERIMENTS AND ANALYSES 354 3.1. Gametophytic incompatibility yields equivalence classes 354 3.2. Deterministic model of incompatibility 356 3.2.1. Determining equivalence classes via crosses 356 3.2.2. Dynamically optimized crossing 356 3.2.2.1. Gain in efficiency 356 3.3. Stochastic model of incompatibility 358 3.3.1. Number of different subdivisions into equivalence classes 358 3.3.2. Probability of the actual data on the basis of a hypothesis H, 359 3.3.3. Estimation of the best hypothesis 360 3.3.3.1. Likelihood ratio of two arbitrary subdivisions H, and Hj 361 3.3.4. An extended single move method for finding a locally optimal hypothesis .... 361 3.3.4.1. Basic formulas in the algorithm 362 3.3.4.2. Extension: amalgamating of two equivalence classes 363 3.3.4.3. Comfortable measures 363 3.3.5. Choice of the most informative forthcoming crosses 364 3.3.5.1. The effect of a single extra cross on likelihood ratio 364 3.3.5.2. Preferable crosses 365 3.3.5.3. ...and how to find them 367 3.3.5.4. Minimum similarity choice 369 3.3.5.5. Maximum lability choice 370 3.3.5.6. Choice of the second partner 372 3.3.5.7. Allocation of resources between populations 373 330 3.3.6. Estimation of seed number probabilities 374 3.3.6.1. Relative seed numbers 375 3.3.6.2. A modified Parzen estimation method 375 3.3.6.3. Return to a discrete distribution 376 3.3.6.4. Estimation of average seed number E [tc | in compatible crosses in a popu- lation 376 3.3.7. Iterative adjusting of equivalence class and seed number probability estimates . 377 3.3.7.1. Choice between end results of different iterations 378 4. RESULTS 379 4.1. Grouping of the sampled ramets into incompatibility classes 379 4.1.1. Crosses 379 4.1.2. No self-fertile clones were found 379 4.1.3. Grouping procedure 380 4.1.4. Crossing recommendations 381 4.1.5. Iterations 383 4.1.6. Estimates of probability distributions 385 4.1.7. Estimates of equivalence classes 387 4.2. Fruiting of the populations in the experimental field 389 4.3. Structure and fruit set in natural populations 393 4.3.1. Observed quantities 393 4.3.2. Population averages according to number of incompatibility classes 394 4.3.3. Population averages according to fruit number class 397 4.3.4. Population averages according to fruiting zone 397 4.3.5. Correlations of population averages 400 4.3.6. Quantities within a population according to equivalence class 402 4.3.7. Correlations of quantities within a population 403 4.3.8. Effect of introducing alien ramets into a population 405 5. DISCUSSION 408 5.1. Breeding system 408 5.2. Dynamically optimized classification 408 5.3. In vigorous natural populations, lack of fruit set is generally due to uniclassic constitution 409 5.4, Rich fruiting presupposes many incompatibility classes 410 5.5. Other factors affecting fruit set 411 5.5.1. Equivalence class size 411 5.5.2. Density, aggregation and patch form 412 5.5.3. Fruiting zone 413 5.6. Vegetative or generative burst 413 5.7. Plant breeding in natural populations 415 6. SUMMARY 417 7. CONCLUSIONS 420 8. ACKNOWLEDGEMENTS 422 9. REFERENCES 423 10. SELOSTUS: INKOMPATIBILITEETTILUOKAT JA MARJONTA MESIMARJAN SUO- MALAISISSA LUONNONESIINTYMISSÄ 427 11. APPENDICES 430 A. Arctic bramble experimental field at Viikki in 1977—80 430 B. Crossing form 431 C. Recording prescriptions, and cross results recorded 432 D. Estimated equivalence classes of populations 433 E. Crossing recommendations 436 F. Details of the crossing technique 437 G. Equivalence relation and equivalence classes 437 H. Number of crosses needed (in a deterministic, dynamically optimized system) to subdivide a population into equivalence classes of incompatibility 437 331 I. A single move method for finding a locally optimal hypothesis 438 1.1. One ramet changes the equivalence class 438 1.2. One equivalence class disappears 439 1.3. A new equivalence class appears 439 1.4. Short-hand version 439 1.4.1. Behaviour of the probability part 439 1.4.2. Short-hand version is more favourable 441 J. Average efficiency of procedures I and II in the selection of a proper ramet as the first crossing partner 442 K. Low similarity to its ‘own’ class is less common in a greater subclass 444 L. Probability of n seeds (taken from a population with three alleles at the incompatibility locus) being of a common incompatibility genotype 446 332 1. Introduction 1.1. Characterization of Rubus arcticus 1.1.1. Taxonomy and distribution The arctic bramble (Rubus arcticus L.) also called arctic raspberry or ‘nectarberry’ (in Swedish ‘äkerbär’, in Finnish ‘mesimarja’) (Tammisola and Ryynänen 1970) belongs to the subgenus Cylactis (Focke) of the genus Rubus. According to Focke (1910), Cylactis comprises of 14 species; three of them (R. arcticus L., R. saxatilis L. and R. humulifoliusC. A. Meyer) have been reported to occur in Finland. The subgenus Cylactis was divided into four series: Arctic!, Saxatiles, Humulifolii and Xanthocar- pi. The series Arctici consists ofR. arcticus(including ssp. acaulis (Michx.) Focke) and R. slellalus Sm. The arctic bramble is a diploid species (2n= 14). In ad- dition ssp. acaulis (on the basis of stomata length, Lars- son 1969), R. humulifolius(Vaarama 1949, 1965) and R. slellalus are diploids, whilst stoneberry (R. saxatilis) is tetraploid (2n =28) (Vaarama 1939, 1954). At the present time, also R. slellalus Sm. is mostly con- sidered to be a subspecies ofR. arcticus, ie. ssp. slellalus (Sm.) Boiv. emend. Hult. (Hulten 1968, 1971, Larsson 1969, 1980a). Larsson (1969) showed that R. slellalus crosses freely with R. arcticus ssp. arcticus, resulting in fully fertile hybrids. HultEn (1968) also reported that hybrid swarms are formed between the three subspecies in areas where the ranges overlap. Ssp. slellalus is pure only in the Aleu- tian islands, where the two other subspecies are lacking. Further evidence for the close relationship between R. slellalus and R. arcticus ssp. arcticus was given by Kal- lio (1975a) from biochemical studies of the aroma com- pounds, and by Larsson (1969) as well as Kotimäki and Hiirsalmi (1979) from cytogenetic studies. Since backcrossing is also easy in both directions, (HultEn 1968, Larsson 1969, 1980a, b, Tammisola un- publ., Kotimäki & Hiirsalmi 1979), some new varieties of this newly cultivated species have been bred via hybrids and backcrosses between these two subspecies. For ex- ample the ‘all-fieldberries’ in Swedish ‘alläkerbär’ by Larsson (1985), in Finnish ‘jalomaarain’ by Hiirsalmi et al. (1986), are a cross between ssp. slellalus, which has provided its desirable robust growth characteristics and ssp. arcticus, which has contributed a certain amount of its unsurpassed aroma (Larsson 1969, 1980b, Kallio et ai. 1980, Ryynänen and Dalman 1983, Hiirsalmi et ai. 1986). The “genuine” arctic bramble, ssp. arcticus, occurs throughout subarctic Eurasia, mainly between 60° and 70° N, but also in Asia in a broader zone, from 50° to 70°. The distribution also extends into the northern parts ofNorth America, ie. Canada and Alaska. Subspe- cies acaulis occurs in the north eastern -most tip of Asia from the shores of the Bering Straits, across Alaska and in subarctic Canada. Ssp. slellalus has a narrow distri- bution range in NW Alaska, the Aleutian Islands and into NE Asia on the shores of the Bering Straits, roughly where the distributions of the other two subspecies coincide (Fig. 1; Hulten 1968, 1971, Larsson 1969). The three subspecies differ in many characters. Ssp. acaulis is tiny and its stem has only one flower, with rather long petals. Ssp. slellalus is considerably larger (40 cm) than ssp. arcticus (15 —25 cm. Figs. 2 and 3), and also has much bigger berries. Its leaves are clearly different in morphology from those ofssp. arcticus: they are more roundish, three-lobed but seldom if ever divided into sep- arate leaflets. In addition, it is more resistant to frost, drought and diseases. Having a more ‘weedy’ growth habit than ssp. arcticus, it also competes better with the other species in a cultivated field. In Finland, only ssp. arcticus occurs. In the records from the early decades of this century, the arctic bram- ble has been reported more or less frequently from every province of Finland. In the southwestern islands, ie. from Aland, it is however almost totally lacking (Vaarama 1965). Upon reviewing these records, Saastamoinen (1930) concluded that in the western coastal and through the middle parts of Finland there clearly exists an optimal zone, in which the arctic bramble occurs frequently and sets fruit regularly and in abundance (Fig. 5). A more cur- rent picture of its occurrence in different parts of Fin- land, at least in forests and in swampy areas, may be ob- tained from Fig. 4. 1.1.2. Aroma The berry of the arctic bramble is used as a foodstuff in several ways. It may be used fresh or frozen, in jam or in liqueur. The best-known product is probably the arc- tic bramble liqueur (Mesimarja), which is currently so popular that the capacity of the production cannot meet the demand. The arctic bramble berry is in particular sought after for its unique aroma, which is clearly different from that of other berries (Saastamoinen 1930, Larsson 1969, 1980b, Ryynänen 1973, Kallio 1975b). Linnaeus (1762), who himself succeeded in cultivating the arctic bramble, was well aware of the delicious flavour of its fruits, which he considered as “de smakeligaste af all frukt ... i hela Europa” [the most delicious of all fruits ... in whole Eu- rope]. The arctic bramble fruits contain more than two hun- dred volatile aroma compounds (Kallio 1976). More than sixty of them have been identified, comprising about 333 334 Fig. I. General distribution of R. arctwus. a) ssp. arcticus (principally in subarctic Eurasia) and ssp. acaulis (in North America) b) ssp. stellatus (after Hulten 1971). Fig. 2. Arctic bramble samples growing in pots in a greenhouse at Viikki, at autumn 1976. 90 per cent of the total aroma concentrate. The most abundant compound among them is ‘mesifurane’, ie. 2,5-dimethyl-4-methoxy-2,3-dihydro-3-furanone. In or- ganoleptic analyses, this compound proved to be primarily responsible for the flavour in the press juice of the arctic bramble (Kallio and Honkanen 1975). It had been iden- tified earlier in canned alphonso mango. In the measurements of Pyysalo et ai. (1977), it expressed ex- tremely high odour value. A closely related compound ‘hydroxymesifurane’ ie. 2,5-dimethyl-4-hydroxy-2,3,- dihydro-3-furanone is also found in the arctic bram- ble thoughonly in small amounts. This latter compound has been reported earlier in pineapple, beef, strawberry, roasted filberts and almonds, popcorn, and heated ma- ple syrup. Since, however, the threshold value of the lat- ter compound is low, Kallio (1976) concludes that these two furane derivatives form the basic odour of the arctic bramble. No research appears to have been done on the aroma constituents in ssp. acaulis. According to the informa- tion given to her from a Canadian research station, Lars- son (1969) supposed that its fruits lacked the arclicus- aroma. Vaarama (1951), on the basis of information available to him (however not specified), believed that the aroma should greatly resemble that of the arctic bram- ble. Having tasted the berries in nature, Y. L. A. Mäki- nen in a letter comments that they could not be separat- ed from the arctic bramble by aroma. The ‘Alaskan raspberry’, ssp. slellalus, being tested orally by Larsson (1969), should however have a ‘pi- quant, smoky but weak’ arclicus-aroma. This consider- ation was confirmed by Kallio (1975 a). He found out that the aroma compounds of ssp. slellalus are qualita- tively very similar to those of ssp. arclicus: each of the compounds identified had also been found in ssp. arcti- cus. However, the total amount of volatiles in ssp. slel- lalus is only from one fourth to one third of that in ssp. arclicus. And the content of mesifurane (the most impor- tant component for the flavour) is from 19 to 44 per cent. In the hybrid between the two subspecies (ssp. stella- lus x ssp. arclicus), the quantity and quality of the aro- ma were as a rule intermediate between parents, though the aroma spectrum showed large variation between different samples (Kallio et ai. 1980). 1.13. Habitats In Finland the arctic bramble occurs both in natural and in lightly or marginally culti- vated habitats. As natural ones may be clas- sified its occurrences on the shores and water- sides, as well as in swampy spruce forests and in northern, well-lit groves. Arctic bramble Fig. 3. A ramet of a triploid arctic bramble clone (population 1 16A) at Maivajoki, Kuusamo, 30lh of July 1976. Notice the leaf morphology: leaflets are more roundish, thick and less lobed than in an ordinary diploid, and often even slightly deformed. 335 seems to be a weak competitor and will easily be replaced by other species in too crowded or too deeply shadowed plant communities (Saastamoinen 1930, Salminen 1948, Ervi et ai. 1955; cf. Elveland 1983). Hence, its nat- ural habitats are capable of retaining their open nature, without too much competition for light and space exerted by taller plant spe- cies. In N Finland, arctic bramble is commonly encountered along the stony banks of rivers and brooks. There the vegetation is probably in part controlled (and also occa- sional spots of bare ground revealed) by ice and floods in spring. (Generative) reproduction in arctic bramble is known to require such spots, since the germinated seeds have been found only on patches of bare ground (Saastamoinen 1930). Reindeers, sheep or other grazing animals may also have a role in keeping theriverside plant communities open enough to permanent growth ofR. arc- ticus. On the sea and lake shores, there often exists a zone of shrubs or low forest, generally dominated by alder spe- cies (Alnus glutinosa (L.) Gaertn. or A. incana (L.) Moench.). On the margins of these zones, vigorous colo- nies of R. arcticus'art often to be found. While arctic Fig. 4. Frequency of occurrence of arctic bramble in forests and swampy areas in Finland, according to the third transectional evaluation of Finnish forests (After Vaarama 1965, modi- fied). Fig. 5. Frequency of occurrence and fruit set of arctic bramble in Finland, before and up to the year 1930. 1 = generally very common and setting fruit richly (optimal zone), II = common and usual- ly setting fruit, HI = fairly common but sets fruit only occasionally, IV = rare (After Saastamoinen 1930, modified). 336 bramble shoots are often seen also deeper in the alder zone, they are as a rule weaker there or even have ceased flowering altogether (Saastamoinen 1930, Ervi et al. 1955, Tammisola unpubl.). The nutritional status of the ground in the arctic bram- ble stands usually exeeds the level reported from other representative areas and on average, the levels of phos- phorus or potassium are higher (Ervi et al. 1955). The high level of potassium may be due to the burning prac- tices (see beneath). The burnt ashes from forest have been used in some instances as a fertilizer for the arctic bram- ble, with reportedly good results (Ervi et al. 1955). Ac- cording to Vaarama (1951), R. arcticus grows well on all kind of garden humus, provided it is not rich in calcium. Salminen (1948) and Ryynänen (1971, 1973) stated that arctic bramble is not restricted by soil pH. The arctic bramble has considerably benefitted from the ancient, minimal cultural practices of man (Linkola 1916). Clearing and burning-over of woodland was exer- cised for centuries as a regular procedure in agriculture. This provided the arctic bramble with bare ground for the germination of its seeds as well as for vegetative spread (cf. Flinn and Wein 1977). R. arcticus often flourished on these “kaski” (“slash and burn”) fields, which were cultivated for a couple of years and then abandoned or transferred to cattle grazing areas (Saastamoinen 1930). The early cultural practices of man created still favoura- ble environment for the arctic bramble. Occasional ploughing since it was still minimal cultivation and did not affect the ground too deeply also inhibited the com- petition whereby the arctic bramble profited. R. arcticus thrived particularly well on the banks of ditches between the field strips. By mowing down his meadows regular- ly, man kept the vegetation low enough, and prevented bushes or tall herbs from occupying them. Thus the meadows remained rather unchanged for long periods of time. If these meadows were also used for grazing animals, which further controlled the vegetation and via tram- pling with their hoofs created bare patches of land, this type of management also produced a favourable en- vironment for the germination of the seeds. As these meadows age, the arctic bramble diminishes in numbers and in vitality (Saastamoinen 1930). Since agricultural practices nowadays have altered radically, most of the above mentioned lightly cultivated habitats are fast disappear- ing. Due to subsoil drainage, open ditches are rare in the fields. Burning is only seldom used in farming practice today. Ploughing is far more intensive, tilling the ground to a much greater depth, which practically destroys the possibilities of shoot regeneration from root fragments of the arctic bramble. The effect of modern, heavy ploughing was readily ob- servable when a colleague accidentally ploughed up my first large collection of Finnish R. arcticus strains in the autumn of 1974. The next spring only a few miserable shoots could be found in the field. In forestry, the modern, strict control of forest fires has caused a great decrease in arc- tic bramble. In former times, forest fires caused by eg. lightening were quite common, and regularly altered the forest ecosystems thus providing the arctic bramble with areas favourable for growth. Fire has, however, still been used to a certain degree in forest management in Finland especially during 1925—35 and 1955—65. In the future, controlled burn- ing may be used more, for its preventive effect on Fomitopsis annosa (Fr.) Karst., the serious decay caus- ing fungus of Norway spruce (Anon. 1980). Modern for- estry management techniques occasionally create suita- ble conditions for the arctic bramble. Open areas gener- ated by patchwise clear-cutting maygive the arctic bramble the possibility to flourish for a period of some years. In addition, the wayside banks of the lumber roads built in the forests often temporarily provide the arctic bramble with suitable habitats. 1.1.4. Spread through growth and reproduction In this paper, with regard to reproduction, the termi- nology of Harper (1977, 1978) has been adopted inas- much as the term ‘vegetative reproduction’ has been replaced by ‘growth’ (see Tammisola 1986). R. arcticus is able to spread vegetatively through its root system. In light soils, the bulk of its roots reach to the depth of 30 cm, and some parts even over 40 cm. In favourable conditions, the rhizomes may extend quite rapidly in a garden the diameter of a clone often increases by about 0.5 m per year, oc- casionally even more (Saastamoinen 1930, Ervi et ai. 1955, Ryynänen 1973, Tammisola unpubl.). In addition, the roots have the high ability to develop adventitious buds, especially after being injured. Thus, according to old records (Saastamoinen 1930, Tammisola un- pubi.), the arctic bramble was able to be ful- ly productive a couple of years following a slight ploughing or burning, and sometimes constituted a continuous under crop beneath eg. a cereal crop which had been previously sown. In rhizomatous plants, a very great propor- tion of the biomass may be in the subterrane- an organs. Figures concerning arctic bramble are, unfortunately, lacking. In R. chamaemo- 337 rus L., Wallen (1986) reports 98 per cent of the biomass to be subterranean. Saastamoinen (1930) speculates that the sudden “appearance” of dense and richly flowering stands of R. arcticus in areas where the forest has been cut down, is based on vegetative spread. The arctic bramble should have been present there already before the clearing, though only as a minor probably weak in vitality, even sterile component of the underlayer. Ervi et al. (1955) even specu- late that the arctic bramble, and its serious rival Rumex acetosellaL., may “hide” under ground for several years, being present solely as resting rhizomes. Such a “disappearance” for some years via dormancy in the soil have also been reported or proposed for some oth- er perennial plants (Tamm 1972, Tammisola 1986). The cutting down of the large trees will abruptly alter the competition between spe- cies, especially the competition for light and nutrients, and the arctic bramble, because of its extensiveroot system, may be able to take advantage of this situation better than most of its rivals. Thus it may even dominate the vegetation for a couple of years, before the species with a stronger competitive ability suc- ceed in taking over. One should, however, note that minor dis- turbances of the environment by man, which are advantageous for the vegetative spread of the arctic bramble, may also be conductive for the germination of its seeds. Saastamoinen (1930) noticed that the seeds may germinate successfully only on bare or almost bare ground, which is a characteristic of weak com- petitors in general (Tammisola 1986). There may be a heavy distribution of R. arcticus seeds in the ground where it formerly flourished. Due to its low germination (usually less than 40 %) (Ervi et al. 1955, Larsson 1957, Ryynänen 1973) and to the thick and hard seed coat, the arctic bramble may have adapted to retain majority of its seeds unger- minated in the ground for long times, wait- ing for the occasionally suitable conditions for germination (Larsson 1955, Vaarama 1965, Tammisola 1981, 1986). Hence, without a more detailed inquiry into the stand in question, one can not conclude which of these explanations for sudden ap- pearance is correct. In a pilot study (Tammi- sola 1981), evidence for the “vegetative burst” explanation was recorded in one stand: there the arctic bramble shoots from both sides of a lumber road were most likely all from the same clone. Along riverside habitats, the seeds of R. arc- ticus may also be dispersed by spring floods (Saastamoinen 1930). Generally they are however distributed by animals eg. birds. In the North Savo Experimental Station, birds especially thrush ( Turdus) species were frequently attacking the arctic bramble plan- tations (Ryynänen 1973), while in my ex- perimental fields at Viikki they were hardly at all interested in arctic bramble fruits. This difference in behaviour may be due to the differ- ent abundance of arctic bramble berries in the two dis- tricts in question. In North Savo, arctic bramble fruits are still fairly common in nature, while on the South coast, they will be found only occasionally. Predators often con- centrate on the prey species which are most common analogouslybirds may tend to be disinterested in arctic bramble berries in the districts where as a rule only a few are produced. In nature, the seeds of R. arcticus will not germinate before the next spring. When al- lowed to dry out, they are known to require a prolonged frost treatment or a treatment with concentrated sulphuric acid before they are able to germinate (Saastamoinen 1930, Larsson 1957, Ryynänen 1973). However, if the seeds from the very beginning are kept in continuously moist conditions, no frost is needed and the seeds will germinate after sev- en months of stratification near freezing point (Ervi et al. 1955, Larsson 1969). The aerial shoots of the arctic bramble are annual. It overwinters via rhizome buds. These are usually situated near the soil level, just below the litter, and are well developed thus they grow rapidly in the spring (Saastamoinen 1930, Zeller 1964). 338 2 1.1.5. Pollination R. arcticus has hermaphroditic flowers and requires insect pollination. Since the filaments of the flowers are thick and curved very close- ly upon the stigmas, the nectaries are only ac- cessible to insects that are sufficiently strong (Fig. 6). It has been shown that in experiments with excluders (cages covered with a net with small meshes), small-sized insects are of little importance in the pollination of the arctic bramble (Ryynänen 1973). In practice, by far the most important pol- linators of R. arcticus are bumble bees and Fig. 6. Flower morphology of R. arcticus. a) In a cross-section of the flower, one can see that the rigid, flattened filaments are in several tight whorls and curved to the centre. Thus, they prevent the entrance to the nectars, and only a few insects, such as bees and bumble bees, or those with a long and thin proboscis, eg. Lepidoptera and Doitchopus flies, can penetrate. b) Being perigynous, the flowers can readily be emasculated by circumcising at the hypanthium with a razor blade. Stamens, sepals and petals will be removed as a common cone, leaving the gynoecium of the bud naked. 339 honey bees (Warming 1886, Poppius 1903, Si- ten 1906, Saastamoinen 1930, Ryynänen 1973, Teräs 1985b). Other insects are only occasionally seem on its flowers, eg. butter- flies. Various dipteras are also sometimes present, as well as beetles and ants. General- ly, though, they have no entrance to the fully concealed nectar, and stigmas, except on damaged flowers. In addition, the pollen- carrying capacity of these insects on their body surface is poor compared to that of bumble- bees and honey bees (disregarding even the pollen in the ‘baskets’, since it has been moistened with nectar and is no more capa- ble of pollinating) (Free and Williams 1972, Kendall and Solomon 1973, Hippa and Koponen 1976). The same applies to the visit- ing speed and activity especially bumble- bees are very active, even at low temperatures and bad weather. Thripsare often found in the flowers of the arctic bram- ble, eg. Thrips major Uz., Frankliniella intonsa Tryb. and Taeniolhrips vulgatissimus Hal. (Ryynänen 1973). They are, however, small, relatively smooth in body surface and especially in pre-adult stages poorly mobile from one flower to another. In addition, the effect of their suc- tion is to ruin the styles and ovaries of the plant, and sub- sequently also the pollen. (Tammisola unpubl.). There- fore, even if the thrips may sometimes be numerous, their role as pollinators in the arctic bramble can at most be considered negligible. Similar conclusions were drawn by Hippa and Koponen (1976) on cloudberry (/?. chamaemo- rus L.). In R. arcticus plots in an orchard in SE Fin- land (Karila, Mikkeli), Teräs (1985b) studied food plants and flower visits by bumble bees and cuckoo bumble bees. Arctic bramble was visited by 9 bumble bee species. Of these, three were rare visitors, while six species were regu- larly found on arctic bramble: Bombus soroeensis (F.), B. pascuorum (Scop.), B. pratorum (L.), B. lucorum (L.), B. hypnorum (L.) and B. lapidahus (L.). The role of cuckoo bumble bees (Psithyrus species) proved to be negligible as pollinators of R. arcticus in the study area. Out of their 1900 flower visits, they only visited arctic bramble once. In comparison to the other 100 plant species in that study, the flowervisits to the arctic bramble were evenly distributed amongst the most frequent bumble bee spe- cies present. Noticeably, however, the long-tongued B. hortorum (L.) was not interested in the flowers of the arc- tic bramble, the corolla of which was shorter (4 mm) than that of species from the study area, whose average was 5 mm. Honey bees may be considered as efficient pollinators, since they are highly flower constant, ie. an individual foraging bee will keep visiting exclusively a single plant species during a foraging trip (eg. Percival 1947). Hence, pollen of, say, arctic bramble will not be lost in- effectively on the flowers of other species. Since a honey bee is able to inform otherbees in the hive when a favoura- ble nectar source has been found (Frisch 1959), a forag- ing strategy based on flower constancy is efficient. In contrast the bumble bee forager lacking infor- mation from other bumble bee individuals must settle on an individual foraging strategy, based on her own sam- plings of the available flower resources. This sampling is time consuming but bumble bees have the advantage that they are able to distinguish the already emptied flow- ers devoid of nectar due to their scent, while still in flight. In addition, individual bumble bees follow their own foraging paths or areas, which further diminishes the time during sampling (Heinrich 1976, 1979, Öster and Heinrich 1976, Teräs 1976, Pekkarinen and Teräs 1977, Nousiainen et ai. 1978). An individual bumble bee has been recorded to visit in from one to several plant species on a foraging trip. One of the species, the ‘major’ one, is usually visited by the bee much more frequently than the other, ‘minor’ ones. A queen during her lifetime may switch her major species several times, often according to the changes in the abundance of flowering species. In addition, individual bumble bees can have distinct multiple specializations, often unrelated to that species’ overall preference (Hein- rich 1976, 1979). Regarding the information available to bumble bees, such a ‘majoring’ strategy has been shown to be always better than random foraging. The continued small scale sampling (‘minoring’) is a necessary com- promise required tokeep track of the resources as they change with time (Öster and Heinrich 1976). The “effectual” flower constancy in bumble bees is higher than one would expect, remembering that usually several species are visited during a single foraging trip. Namely, major proportion of the pollen collected by the bumble bee will be from the majored plant species (Pek- karinen and Teräs 1977). Thus, bumble bees may be considered efficient pollinators with regard to their major plant species. Furthermore, the foraging trips are “piece- wise flowerconstant”, ie. they usually consist of a fairly continuous sequence of visits on a single plant species, with occasional switching to a sequence on another spe- cies (Heinrich 1976, 1979). Hence, pollination of a minor plant species may also be more efficient than ex- pected. Unfortunately, however, theresults present- ed by Teräs (1985b) do not reveal, whether arctic bramble was majored by any individu- al bumble bees at any time. The conclusion of the study was that R. arcticus was not amongst the four most often visited plant spe- cies by any Bombus species, except for B. 340 soroeensis. Visits to the arctic bramble by bumble bees of this species, only constituted 8 per cent or less of their total overall flower visits. This small percentage points to the arc- tic bramble as being a minored plant. Taking into consideration the possible specialisation of individual bees within a bumble bee spe- cies, then the bulk of the visits to the arctic bramble may still be exercised by bumble bee individuals majoring the arctic bramble, at least during its principal flowering period in June. This detail may well prove essential con- sidering the pollination efficiency and fruit set in arctic bramble. 1.1.6. Self-sterility Self-sterility has been reported in many species from various subgenera of the genus Rubus, eg. in R. odoratus L. (subgenus Anoplobatus), R. allegheniensis Porter (subg. Eubatus), R. parvifolius L. and sometimes in R. idaeus L. (subg. Idaeobatus) (East 1940, Fryxell 1957, Knight and Keep 1962). Al- though self-sterility predominates in wild Idaeobati, it is of little importance within commercial Idaeobati, appearing only occa- sionally in certain progenies of the red rasp- berry and in hybrids between the red and black raspberries (Knight and Keep 1962, Keep 1972, Redalen 1976). Thus, during its centu- ries of cultivation (at least since 1548, see Vaarama 1965), due to a more or less uncon- scious selection, raspberry has turned into a self-fertile species as has happened with many other cultivated species as well (eg. Schwanitz 1967, Simmonds 1979). Saastamoinen (1930) concluded that the flower morphology of R. arcticus suggests self-pollination: the filaments are curved to the centre and the anthers of the innermost stamens almost touch the stigmas (see Fig. 6). Such a self-pollination mechanism has been reported to be functioning in some self-fertile Rubus cultivars, as well as in the pseudoga- mously apomictic blackberries (subgenus Eu- batus) (Nybom 1985, 1986). Salminen (1948) reported that arctic bram- ble would have self fertile as well as self ster- ile strains, the latter however being more com- mon. According to him, the self fertile strains should even be morphologically clearly iden- tifiable from the self fertile ones (Ervi et al. 1955). Ryynänen (1973), however, criticized the methods applied in these experiments, regarding clonal contamination as very likely. Isolation experiments on separate clones by Larsson (1969) suggested that at least all the studied clones of the Arctici-series were self- sterile. This was largely confirmed in pollina- tion experiments by A. Ryynänen on 16 Finn- ish strains of the arctic bramble (Tammisola and Ryynänen 1970). Twelve of these strains proved clearly self-sterile, in one strain the overall fertility was poor, and in three strains occasional cases of moderate druplet forma- tion were recorded amongst unequivocally self-sterile reactions. Such an occasional druplet formation could be ex- plained by facultative apomixis, or supposing that the strength of the self-sterility is much influenced by the en- vironment. Apomixis, however, has been stated not to occur in diploid Rubus species (Knight and Keep 1962), thoughit is a common phenomenon among polyploid Eu- bati. In a cytological study with R. saxalilis a tetraploid species in the subgenus Cylaclis Czapik (1981) regard- ed apomixis possible, though there was no direct evidence of successful apomictic reproduction. After all, technical inconsistency may be regarded as the most likely explanation for the drupelet formation (cf. Fowler and Janick 1972). Since in these pilot studies, both the quality of the isolation and the purity of polli- nation in addition to the pollen and clonal sources were not always carefully controlled. Larsson (1969) also reports an exception- al case of a small amount of self-fertility to- wards the end of July in one year. She tried to check this in later years, using only very carefully isolated flowers, but not fruits ever developed. At the Komarov Institute in Leningrad there is reputed to be a self-fertile strain of the arctic bramble (Härdh 1976), but I have not succeeded in obtaining any confirmation of this. Thus, until now there seems to have been no reproducible or confirmed cases of self-fertility in diploid R. arcticus. In pollination experiments using parents and their F,-progeny (Tammisola and RyynA- 341 nen 1970), it was shown that the self-sterility is due to an oppositional type of an incompat- ibility system, controlled by a single gene locus and many alleles. The system was further shown to obey a sporophytic-gametophytic control ofpollen tube growth. This means that the arctic bramble has a self-incompatibility system of the Oenothera-type, which is fairly widely distributed in the family Rosaceae. Typical of this system, the self-incompatibil- ity (at least partially) no longer held true at the tetraploid level (Tammisola and Ryynä- nen 1970), though Larsson (1969) reported the autotetraploids of R. arcticus to be self- sterile. 1.2. Variation of fruit set in nature Saastamoinen (1930) divided Finland into zones on the basis of the occurrence and productivity of the arctic bramble (Fig. 5). Across Central Finland, broadening towards the North in the Eastern coastal regions of the Gulf of Bothnia, should be its optimal zone. Though the arctic bramble could be found also to the South and to the North of that zone, in these areas it set fruit only occasion- ally and in small amounts, despite locally rich flowering. Saastamoinen proposed several reasons for the recorded variation in occurrence and fruit set. The arctic bramble may be too continental for thriving in certain areas. In her opinion, general temperature maps seemed to correlate well with the zones introduced by her. M. J. Kotilainen (Saastamoinen 1930) suggested that too hot summers make the flowers to dry. Thick snow cover might be profitable: it pre- vents an early start of growth, thus protect- ing the flowers against frost damage in spring the best flowering period of the arctic bramble. In Southern Finland, intensivefarm- ing may have harmed it, by robbing it of suita- ble habitats; in addition, heavy clay soils are often encountered in the South a soil type apparently poorly suited for R. arcticus. The overall productivity of arctic bramble in Finland is well known to vary very much from one year to the next. In essence, this var- iation must be due to different weather con- ditions. In addition Saastamoinen (1930) no- ticed that there were also great differences in fruit set inside the zones. These local differ- ences were attributed by her to differences in moistureand illumination.For good fruit set, adequate moisture is required. According to her, light, however, should not be a limiting factor, since even in shaded Alnus thickets, there is plenty of light in spring and early sum- mer. Ervi et al. (1955) considered that arctic bramble might even favour slight shadowing. Ervi et al. (1955) suggested that outside of the optimal zone, there may be more severe competition from the rich natural flora, as well as scarcity of habitats favourable to the arctic bramble. They also considered the pos- sibility that the zonation might be based on the presence of mycorrhizalfungi. This hy- pothesis, however, was rejected on the basis of the results in their study. They further point to the possible role of micro nutrients. Using semi-quantitative analysis, no differences, however, could be recorded in the trace ele- ments from fruiting versus non-fruiting shoots. Comparing the nutrients in the soil, they found however clearly more phosphorus, copper, zinc and manganese in heavily productive areas than in those areas which gave low yields. Further explanations based on microclimate have been given. Kotilainen (1949) supposed that in spring and early summer, less frost damage occurs near the lake districts (cf. Solantie 1976), which results in greater arc- tic bramble yields in these areas. Some evi- dence is given by Ervi et al. (1955) that in a plastic greenhouse, more fruits are produced. In a plastic greenhouse, Hursalmi (1971) ob- tained best yields in a mist compartment. He thus concluded that the weak berry produc- tion of the arctic bramble in Southern and South-Western Finland could be due to low relative humidity of the air in spring (which he believed to prevent the dehiscence of the anthers). This generalization to encompass 342 natural conditions may, however, be too far- reaching, taking into consideration the ‘exot- ic’ conditions prevailing in the greenhouse. For instance, the temperature was not con- trolled and accordingly it was much higher in the plastic greenhouse than in nature. Ervi et ai. (1955) note that the distribution of thepollinating insects could be of great im- portance concerning fertilization. As was al- ready stated above, only honey bees and bum- ble bees are worthy of consideration. The honey bee only occurs in Finland in domestic hives. About 80 per cent of the bee keepers in Finland are situated in five South- ern or Southwestern provinces ie. Kymi, Uusimaa, Turku & Pori, Häme and Vaasa (Anon. 1976, Varis 1981). Bee keeping is rarely practised in other parts of the country. The approximate northern limit for bee keep- ing is from Kainuu (say, Suomussalmi, 65° N) in the East to Rovaniemi (66.5° N) in the West. The effect of a bee hive is, however, only local, since the normal flying range of the honey bee may be less than 1 km (Rib- bands 1951, Beutler 1954), though even dis- tances as much as 10 km have been reported (Frisch 1959). The overall distribution of honey bee colonies in Finland is quite low, only about 0.09 colonies per square kilometre (Varis 1981). The densities of bumble bees are difficult to measure and vary greatly both annually and during the season, as well as according to lo- cality, habitat or species. Hence, no overall estimates for bumble bees in different parts of Finland are available (Pekkarinen et ai. 1981, Teräs 1983). However, on the basis of distribution information (Pekkarinen and Teräs 1977, Pekkarinen et ai. 1981) and sporadic studies (Teräs 1983, 1985a), a rough general overview can be formed. Hence, bum- ble bees seem to be commonly found all over Finland. In N Finland, the species are very different from those in S Finland. Yet there is no indication that there should be a differ- ence in the overall density of bumble bees be- tween the North and the South. Only in Lapland the densities of the relevant bumble bee species are known to be essentially lower. The local abundance of bumble bee colo- nies is likely in part to be dependent on the availability of favourable nectar and pollen sources early in spring. Modern, intensive farming practices have greatly decreased the occurrence of many such plant species, eg. wil- lows, around and in the fields (Pekkarinen et ai. 1981, Teräs 1985a, b). Bumble bees, how- ever, are known to search for nectar and pollen at distances of more than a kilometre (Teräs 1983). Though comparable records are missing, there seems to be no reason to suppose that the northern bumble bee species should dis- play an entirely different type of flower preference. Thus, supported by scattered records from nature, it can be concluded that the northern bumble bees keep visiting R. arc- ticus as well. Hence, variation in densities of pollinating insects can not generally explain differences in the fruit set, except perhaps in Lapland. Zeller (1964) studied the ontogenetic de- tails in the overwintering buds of the arctic bramble. In her material, strains from the ‘op- timal’ zone had a smaller number of flower primordia (1—3) than either the strains from South or from North Finland (3 —6). Another explanation also based on strain differences in different localities, had been proposed by M.Sc. (Agr.) M. Salminen (Ervi et ai. 1955, Vaarama 1965). He believed that self-fertile strains of the arctic bramble would occur frequently in Central and occasionally in Northern Finland, while they would sel- domly be encountered in South Finland. Hence, the zonal and also local differ- ences in fruit set reported by Saastamoinen (1930) could be attributed to the distribution of self-fertile strains. In Southern Finland, due to the self-sterility of the prevailing strains, almost no berries would be produced. Hence, Ervi et al. (1955) propose that in further studies it should be made clear whether the strains in the ‘optimal’ zone differ from those 343 outside. Tammisola and Ryynänen (1970), using material consisting of only 16 strains, reported evidence contradictory to Salminen’s hypothesis which has been presented above. Only a few strains showed signs of possible disturbances in self-sterility, and contrary to Salminen’s hypothesis, these were from the ‘poorly producing’ zones instead of from the more richly producing ones. Thus, from the ‘optimal’ zone, only self-sterile strains were found. Salminen (1948) and later on also Larsson (1968) and Y. L. A. Mäkinen (see Tammisola and Ryynänen 1970) have suggested that cer- tain cases of non-fruiting in natural occur- rences of arctic bramble might be explained by uniclonality and self-sterility. To formu- late this hypothesis more generally: the clonal composition ofpopulations is proposed as an explanation to the ‘odd’ variation in fruit set in naturalR. arcticus populations in Finland. 1.3. Hypotheses and study plan In order to ascertain the relevance of these last mentioned propositions, the present study was started. To begin with, the following hy- potheses were formulated. 1) All or practically all clones are postulat- ed to be self-sterile. If any (exceptional) self- fertile clones might occur, they are predicted to be distributed in a non-systematic way in relation to the productivity zones. 2) The local differences in fruit set are primarily due to a different availability ofsuc- cessful pollen. If there is a sufficient number of foraging bumble bees or honey bees, the availability of successful pollen will primarily depend on the numberandspatial pattern of the ‘equivalence classes of incompatibility’ in the population (Tammisola and Ryynänen 1970,Tammisola 1981). Hence, vigorous, richly flowering populations with no fruit set should general- ly be regarded as containing a single equiva- lence class. On the contrary, richly fruiting populations should as a rule display a mixture of many incompatibility classes. In theory, two classes would suffice, provided they were even in numbers and intimately intermixed. Vigorous populations with a poor fruit set should usually consist of only two or few equivalence classes. These classes should ei- ther be very uneven in numbers or be well separated spatially. In Lapland, and perhaps in intensively cul- tivated areas outside the bee keeping area, there may occasionally also occur local lack of effective pollinating insects. One should, however, notice that in North Finland, early summer frosts are more common than in Southern Finland. Hence, loss of fruit crop may also be due to climatic factors (Vaara ma 1965). To check these hypotheses, the following general plan of study was initiated. Samples should be taken from arctic bramble popula- tions in different parts of Finland. These sam- ples should be analyzed to yield the number and spatial pattern of the equivalence classes of incompatibility in each population. Self- fertile clones should be looked for in a large number of populations. This data should be compared to thefruit set information from the populations in question. In this study, generally only vital popula- tions ie. vigorous both in growth and in flowering were considered. In each of arc- tic bramble productivity zones (Fig. 5, Saastamoinen 1930), fruiting as well as non- fruiting populations were included in the study. Thus it would be possible to reveal any zonal differences in the equivalence class structure of eg. non-fruiting populations, as well as in the occurrence of (speculative) self- fertile clones. 344 2. Material and methods 2.1. Sampling of populations During the first four months of 1976, a questionnaire was distributed via mass com- munication media in Finland, asking for in- formation about arctic bramble. Inquiries were made regarding in particular vegetative- ly vigorous and richly flowering but yet fruit- less populations. During 1976—1977, more than 300 persons responded to the question- naire. On the basis of this accumulated infor- mation, the populations for the present study were chosen. Adjusting the study plan (see 1.3.), in each of the four “productivity zones” of arctic bramble (Fig. 5, Saastamoinen 1930), both non-fruiting, poorly fruiting and richly fruiting populations were to be inves- tigated. Within each zone and in each fruit- ing class, these populations were acquired through random sampling. During the summer 1976, collection tours to some of these populations were made. In spite of careful checking of the information in advance with the correspondents, a few of the populations first occurring in the collec- tion scheme proved in situ to differ too much from the reported fruiting class, thus render- ing modification of the collection scheme necessary. For practical reasons, the popula- tions which were visited, were as a rule includ- ed in the collection scheme, but now represent- ed their actual fruiting class. In 1977, a revised collection scheme was made: the still missing populations of each fruiting class were selected at random. Also in 1977, a couple of origi- nally selected populations had to be replaced by other ones, due to their wrong fruiting class or insubstantial growth area. The smallest growth areas were already excluded prior to random sampling the minimum acceptable size of the longest diameter was set to approx- imately 10 metres. A total of 29 populations were taken (Fig. 7, Tab. 1). 2.2. Sampling of shoots From each population, several shoots were sampled at random. The rootstocks of these shoots were dug up and sent for planting at our experimental field in Viikki, Helsinki. On the basis of this ramet sample (for the termi- nology, see Tammisola 1986), the equivalence class pattern of the population could be ana- lyzed. 2.2.1. Number of ramets to be taken in a population In order to study as many as 29 popula- tions, the numberof ramets from each popu- lation cannot be too exhaustive. This number should, however, be large enough to reveal at least “essential” differences between popula- tions, with regard to the number and spatial pattern of their contained equivalence classes of incompatibility (see 3.1.). Thirty ramets were considered to be enough, on the basis of following reasons. I. Postulating m equally frequent equiva- lence classes in (an infinitely) large population of shoots (cf. Tammisola 1986, ‘Richness’), theproblem concerning the number of shoots to be taken can be studied via the theory of ‘occupancy’ (see Riordan 1958, p. 90). Taking shoots from the population(ie. from an array of m equivalence classes) is analogous to putting objects into m cells, each with equal probability. Combining his results (Riordan 1958, p. 90—91), probability of no 345 346 Fig. 7. Arctic bramble populations chosen for the present study. The populations which remained (almost) totally unanalyzed into equivalence classes, have been denoted by open symbols and small type. Fruit set is indicated: dot = rich, asterisk = poor, star with five points = none. Table 1. Arctic bramble populations chosen Popul. Fruit Prod. Sampled Popul. Longitude Latitude label 1 set2 zone3 area con- ■ ' E ° ' N [m xm] tinues 4 002 A 0 1 Bxl4 no 28 06 62 54 004 A 2 I 1x25 int. 25 58 62 16 0048 l a I 4x26 no 25 58 62 16 010 A 2» 111 9x14 yes 24 15 60 09 021 A 2 I 2x21 yes 24 29 65 52 030 A 1 I Bx9 int. 29 15 63 13 032 C 2 II 9xlB no 23 38 62 17 (037 A) 2 I 12x14 yes 25 02 65 02 0388 0 IV 13x14 no 25 26 61 10 039 A 1 111 4x21 no 24 40 6104 042 A 2 II 7xlo yes 27 45 64 16 (051 A) 0 111 8x32 int. 26 44 68 55 (063 A) 2 111 9x13 yes 27 32 68 40 1066AJ 0 111 BxlB yes 29 22 67 48 [O66D] 0 111 5x25 yes 29 22 67 48 075 A 2 IV 3 X 25 no 26 57 69 44 078 A 1 I Bxl3 no 27 49 63 26 087 A 2 II 10x11 int. 25 38 66 35 0898 0 II 4xlB yes 25 07 67 03 (090 A) 2 I 5x23 int. 25 24 64 51 099 A 0* 111 6x21 no 23 35 60 18 102 E 0 I 5 x 26 no 22 19 62 59 1158 0 I 11x14 no 28 01 61 59 116 A 0 111 10x14 yes 29 50 66 01 186 A 2 111 9xlo int. 26 52 60 29 190 A 0 IV 5x21 int. 24 55 6149 (204D) 2 111 2x23 int. 28 44 65 55 212 A 2 111 Bxl3 yes 2154 6138 1304A] 0 II 15x20 int. 25 29 66 24 1 Due to their weak condition in the experimental field, populations enclosed in square brackets [ ] remained totally (and those in parentheses (), almost totally) unanalyzed into equivalence classes. 2 'o' = none, 'l' = poor, '2' = rich; representing less than 1, 1—l5, and more than 15 'full' berries (ie. containing at least 10 drupelets) per a totally covered (100%) m 2 of arctic bramble vegetation. ' See Fig. 5. 4 ‘yes’ = arctic bramble occurs almost continuously in at least about two times of the area sampled;‘int.’ = popula- tion continues but intermittent, ie. with at least about a 20 m break. • Date of collection was too early for reliable in situ records of fruit set. Hence, classification is primarily due to the previous and posterior reports of the correspondents. empty cells remaining after putting n objects at random into m cells will be m!(1) P (n,m;m) = —--8(11,111), m" where S (n,m) is a Stirling number of the second kind. This formula will therefore also tell the probabilityof get- ting all m equivalence classes represented in a sample of n shoots from the population. These Stirling numbers have been tabulated, eg. in Riordan (1958) up to 10 and in David et al. (1966) up to 25. Unfortunately, there were no tables that could have been used for a sample size greater than 25. Therefore, an expression for the calculation of the desired probabil- ity was derived (2) P (n,m;m) = m-1 m i1- £ (-1)- 1 • {.) • (1 )". i -1 i m Probabilities up to the sample size of 60, which were calculated utilizing this expression, are presented in Ta- ble 2. By using a sample size of 30, it is almost certain to obtain all of the equivalence class- es present in the population, whenever the population contains at most 6 different equivalence classes. The risk of loosing any 347 of the equivalence classes will then be less than 3 per cent. Using a sample size of 15, therisk of loosing equivalence classes were consider- ably greater: 36 per cent for populations with 6 classes. At a risk of 5 per cent, all classes can be guaranteed in a sample only in popu- lations with at most 4 classes. At this risk lev- el, a sample size of 60 would suffice for popu- lations containing at most 11 classes. Since, however, the amount of labour re- quired in the analyses will increase very rapidly with the number of ramets to be studied (see 3.), a sample size of 60 is far too large to be used in the study. A sample size of 30 is still manageable, though only via the careful op- timization of procedures as applied in this study. It can be shown that P (n,m;s) (ie. the probability of getting exactly s equivalence classes, out of a total m in the population, into a sample of size n) can be traced back to the expression (2), viz. m s (3) P (n,m;s) = ( ) ■ ()n • P (n,s;s) m —s m With reference to a population containing m= II equivalence classes, the probability distribution of the number of equivalence classes occurring in a sample of size n=3o, is given below: N:o of classes I—6 pooled 7 8 9 10 11 Probability .000006 .0004 .01 .10 .40 .49 This example shows that almost always, nearly all of the classes will be contained in the sample just as could be supposed on the basis of the high (10.4) number of expected classes (Table 2). According to Riordan (1958, p. 101), the expectation of the number of equivalence classes, s, occurring in a sample will be (4) E {s] =m•[l (1 —— m These expected numbers of revealed classes have also been presented in Table 2. 11. Postulating m unequally frequent equivalence classes in (an infinitely) large population of shoots, the problem of sample size turns out to be much more difficult. It is, however, still simple to study the most se- vere type of under-representation in a sample, Table 2. Probability of acquiring, into a sample containingn ramets, every one of the m equivalence classes present in the population, provided all the classes are equally frequent. Sample size (n) 5 10 15 30 60 pa Eb pa Eb pa Eb Pa Eb pa Eb m = 2 .94 1.9 .998 2.01.00 3 .62 2.6 .95 2.9 .99 2.01.00 2.01.00 2.0 3.01.00 3.01.00 3.0 3.9 .999 4.01.00 4.0 4.8 .99 5.01.00 5.0 5.6 .97 6.01.00 6.0 6.3 .93 6.9 .999 7.0 6.9 .86 7.9 .997 8.0 7.5 .76 8.7 .99 9.0 7.9 .63 9.6 .98 10.0 8.4 .49 10.4 .96 11.0 8.7 .36 11.1 .94 11.9 4 .23 3.1 .78 3.8 .95 5 .038 3.4 .52 4.5 .83 6 .27 5.0 .64 7 .10 5.5 .43.10 5.5 .43 8 .028 5.9 .25 9 .0047 6.2 .12 10 .00036 6.5 .046 II .014 12 .003 13 .0006 9.1 .24 11.8 .90 12.9 .00006 9.4 .15 12.5 .84 13.8 .000003 9.7 .088 13.1 .78 14.8 .046 13.7 .70 15.7 .022 14.2 .62 16.6 .010 14.8 .53 17.4 .004 15.2 .45 18.3 .001 15.7 .36 19.1 14 15 16 17 18 19 20 a Probability P(n,m;m) (see expression 2). b Expectation of the number of equivalence classes in the sample, that is Enjn fsj (see expression 4). 348 which would lead to a critical misinterpreta- tion of the equivalence class structure of the population. That is, if the sample of shoots happens to contain only one equivalence class, in spite of several which occur in the popula- tion, the population will be fatally misclassi- fied, regarding the central problem of the study. Denoting the relative frequencies of the equivalence classes in the populationby p, the probability to miscias- sify the population as uniclassical will be (5) Pmis = p, n + p2 n +...+pm n