Fertility Variation and Gene Diversity in Forest Populations

: Fertility variation, deﬁned as a difference in the ability to create progeny (i


Introduction
Foresters and agriculturalists expect abundant reproductive output for the highest yield and low cost [1][2][3].Plant growth is promoted for different treatments, such as pruning, hormone application, and soil fertilization.However, geneticists focus on the equal or acceptable contribution of individual plants to gene pool, which can produce a genetically high-quality seed crop [4][5][6].High variations in the contribution also could help to achieve a balance using theoretical and practical tools [7][8][9], while large fertility variations were reported in different populations and years in 99 stands and 36 seed orchards of different forest tree species [10][11][12], and in Pinus brutia [13] and Cedrus libani [14].
Fertility, also referred to as fecundity, is the ability to produce progeny in the next generation through reproductive traits.Estimating fertility variations serves a multitude of purposes, such as the estimation of gene diversity in seed orchards, forest genetic resource management, gene conservation, regeneration practices, and evolutional studies.In the realm of plant genetics and breeding, the estimation of fertility variations stands as a significant tool [10][11][12][13][14][15][16][17][18][19].Moreover, it has wide application in the selection, establishment, and management of seed sources [18].One aspect of these applications involves considering gene diversity in order to monitor and increase genetic variation in seed crops [11,17].

Materials and Methods
The published papers were surveyed using the keywords "fertility variation" and other linkage parameters such as "status number", "gene diversity" and "sibling coefficient" from the Web of Science and Google Scholar databases.English papers were selected from the databases.They were combined according to the unpublished experience and knowledge of the authors.

Variations in Female and Male Fertility
Female and male fertilities of the ith individual (denoted as ψ f and ψ m ) were defined as the ability to produce female and male strobili, respectively.This fertility difference was estimated by calculating the relative proportion of female and male strobilus production in relation to the entire population, following the method proposed by Muller-Starck and Ziehe [53].To assess the variations in female and male fertility, also known as female and male gametic fertility variations, the coefficient of variation (CV) for female and male strobilus production was employed, as suggested by Kang and Lindgren [32].The estimation of ψ f and ψ m was carried out using the following equations: In these equations, N represents the census number, f i and m i correspond to the fertilities of female and male ith individuals, and CV f and CV m denote the coefficients of variation in female and male strobilus production, respectively, among individuals in the studied population.
Moreover, the fertility variations among individuals were estimated based on the proportion of cone production (e.g., fruit, conelet, acorn, and berry) within the population.The variations in cone fertility (Ψ C ), with the total contribution representing zygotic parents, was estimated by Kang and Lindgren [29] and by Bilir [30], as expressed by the following equation: Here, Con i represents the cone fertility of the ith individual; CV 2 C stands for the coefficient of variation in total fertility.In this paper, the fertility of the ith individual was estimated by the proportion of cone production in the population.
However, many biotic and abiotic factors can affect this proportion.For instance, a positive and significant correlation was reported between female and male strobili productions in Pinus taeda [54], opposite to that reported for P. eliotti [55], P. sylvestris [56] and P. contorta [57].In addition, positive and significant correlations were found between the numbers of cones and filled seeds in Picea sitchensis [3], P. abies [56] and Pseudotsuga menziesii [2,25].Bhumibhamon [57] reported a positive and significant relation between strobili production and crown volume in Pinus sylvestris [58], similar to Picea abies [59,60] while this correlation was negative in Pinus taeda [54] and P. sylvestris [61].Low correlations were reported between tree height and strobili production in Pinus contorta [57] and Picea abies [62].Tree age was an important factor in seed production in Pinus sylvestris [39,63].The results also indicated the importance of genetical and traditional (i.e., pruning) practices in the proportion [64][65][66].

Total Fertility Variation (Sibling Coefficient)
Sibling coefficient is defined as the probability that sibs occur compared to the situation in which parents have equal fertility.It is a standardized measure that is independent of the census number of parents, and only dependent on how variable their fertility variation is [11].The combined variations in the fertility of both females and males lead to the total fertility variation, designated by the symbol of Ψ and referred to as the sibling coefficient.The total fertility variation (Ψ) can be calculated using the equation by Kang and Lindgren [29]: The total fertility variation (Ψ) can be also calculated using Kang's equation [11]: In Equations ( 3) and (4), N is the census number, f i and m i are the fertilities as female and male parents of the ith individual, respectively.p i is the total fertility as the whole parent of the ith individual, which is the average of female and male gametophytes' contribution to the offspring.
Equations ( 3) and (4) were simplified as follows [13]: If there is no correlation between female and male fertility, total fertility variation (Ψ) is calculated based on the coefficient of variations in female (CV f ) and male (CV m ) fertility by Kang and Lindgren [32], and based on the female fertility (ψ f ) and male fertility (ψ m ) variations recorded by Kang and Lindgren [29] and Bilir [30] as: When equal seed harvesting is imposed in a seed stand population, Formula (6) is then described as Ψ = 0.25ψ m + 0.5, where only male fertility variation remains.This can then be expressed as stated by Kamalakannan et al. [40]: Equations ( 3) and ( 5), delivered by Kang and Lindgren [32] and Bilir et al. [13], were improved under the correlation (r) between female and male fertility.The new equation was implemented for the estimation of total fertility variation (Ψ) [67] as follows: where r is the correlation coefficient between female and male strobilus production in the population.Formula ( 9) is also improved as a theoretical framework for the estimation of total fertility variation (Ψ), as presented by Bilir and Kang [17]: where N is the census number, ψ f is the female fertility, ψ m is the male fertility and f i and m i are the number of female and male strobili of the ith individual, respectively.Fertility differences among population members can be described by the coefficient of variation (CV) in fertility and the size of the sample (n), as stated by Kang [11]: Forests 2023, 14, 2172 5 of 14 When making predictions for objects that are neither juvenile nor characterized by poor flowering, a rough generalized heuristic rule is suggested: Ψ equals 2 (CV in fertility = 100%) for seed orchards and Ψ = 3 (CV = 140%) for natural stands [10,12].According to Equation (10), the Ψ value will be larger than 1.If all individuals contribute equally, then the Ψ equals 1, and the Ψ = 2 when the probability that two individuals share a parent is twice as high compared to when parental fertility is equal across the population [18].A Ψ lower than 2 is an acceptable level in most of the empirical studies.However, it is expected to be 1 in an idealized situation (i.e., under the Hardy-Weinberg equilibrium) where mating is random and individuals are equally fertile in a population without any disruptive circumstances (i.e., under conditions that rule out mutation, migration, genetic drift, and natural selection).
In Equations ( 1), ( 2), ( 5), ( 6) and ( 10), fertilities are related to the coefficient of variation (CV).Forest owners and seed source managers expect equal parental gamete contribution from all parents to decrease the CV value (Figure 1).In orchard A, all parents contribute equally, so there are no fertility variations, and the census number for the orchard is the same as the status number of the orchard crop [11,18].
of variation (CV) in fertility and the size of the sample (n), as stated by Kang [11]: When making predictions for objects that are neither juvenile nor characterized by poor flowering, a rough generalized heuristic rule is suggested:  equals 2 (CV in fertility = 100%) for seed orchards and  = 3 (CV = 140%) for natural stands [10,12].According to Equation ( 10), the  value will be larger than 1.If all individuals contribute equally, then the  equals 1, and the  = 2 when the probability that two individuals share a parent is twice as high compared to when parental fertility is equal across the population [18].A  lower than 2 is an acceptable level in most of the empirical studies.However, it is expected to be 1 in an idealized situation (i.e., under the Hardy-Weinberg equilibrium) where mating is random and individuals are equally fertile in a population without any disruptive circumstances (i.e., under conditions that rule out mutation, migration, genetic drift, and natural selection).
In Equations ( 1), ( 2), ( 5), ( 6) and ( 10), fertilities are related to the coefficient of variation (CV).Forest owners and seed source managers expect equal parental gamete contribution from all parents to decrease the CV value (Figure 1).In orchard A, all parents contribute equally, so there are no fertility variations, and the census number for the orchard is the same as the status number of the orchard crop [11,18].Unlike agricultural crop plants, forest trees have large differences in fertility.Additionally, the reproductive capacity of trees varies greatly depending on their age.In years of good seed production, the variations in fertility smaller among individuals, and the variations are larger in poor years [15,23,30].

Coancestry and Group Coancestry
Coancestry (f, θ) is defined as a quantification of the relatedness between two individuals, representing the probability that genes taken from those individuals are identical by descent (IBD).Synonyms for coancestry include coancestry coefficient, kinship, and consanguinity.
Group coancestry (Θ) is the probability that two individuals are IBD, and this term was introduced by Cockerham in 1967 [43].The group coancestry (Θ) of orchard crops can also be calculated from the contributions of parents (pi) using the formula by Kang [11,64]: Unlike agricultural crop plants, forest trees have large differences in fertility.Additionally, the reproductive capacity of trees varies greatly depending on their age.In years of good seed production, the variations in fertility smaller among individuals, and the variations are larger in poor years [15,23,30].

Coancestry and Group Coancestry
Coancestry (f, θ) is defined as a quantification of the relatedness between two individuals, representing the probability that genes taken from those individuals are identical by descent (IBD).Synonyms for coancestry include coancestry coefficient, kinship, and consanguinity.
Group coancestry (Θ) is the probability that two individuals are IBD, and this term was introduced by Cockerham in 1967 [43].The group coancestry (Θ) of orchard crops can also be calculated from the contributions of parents (p i ) using the formula by Kang [11,64]: If all parents are assumed to be unrelated and non-inbred, all self-coancestry equals 0.5.When they are related to each other (i.e., θ ij ), the group coancestry can be calculated as follows: Forests 2023, 14, 2172 6 of 14 Group coancestry (Θ Ψ ) is estimated by considering parental fertility (p i ) male and female fertility [64] as: Here, N is the census number, f i is the female fertility, m i is the male fertility of the individual i, and p i is the probability that two genes in the offspring come from the same parent i.The accumulation of group coancestry is faster and higher when the fertility variation is large [10,11,33], as shown in Figure 2.

Θ = . ( ) ∑ ∑
Group coancestry () is estimated by considering parental fertility (pi) male fertility [64] as: Here, N is the census number, fi is the female fertility, mi is the male fe individual i, and pi is the probability that two genes in the offspring come fr parent i.The accumulation of group coancestry is faster and higher when the iation is large [10,11,33], as shown in Figure 2.

Gene Diversity and Status Number
Gene diversity (GD) is one of the most important criteria to assess the qu crops and resistance of forest establishment to biotic and abiotic damages warming), ass discussed by Ivetić et al. [65].Gene diversity could be an env friendly solution to biotic (e.g., insect damage) and abiotic (e.g., climate chang based on artificial and natural selections.Gene diversity can be a reflector o quality of seed crop, as well as its commercial value and choosing by the fore Group coancestry represents the probability that two genes in a popula Diversity refers to differences, and gene diversity indicates differences in gen 1-group coancestry () denotes the probability that the genes are non-ident diverse (i.e., gene diversity).

𝐺𝐷 = 1 − Θ
The number of "ideal" trees shows the trees that have the same gene div considered population.The status number (Ns) is a way of expressing grou

Generations
Equal fertility

Female constant
Female and male vary

Gene Diversity and Status Number
Gene diversity (GD) is one of the most important criteria to assess the quality of seed crops and resistance of forest establishment to biotic and abiotic damages (i.e., global warming), ass discussed by Ivetić et al. [65].Gene diversity could be an environmentally friendly solution to biotic (e.g., insect damage) and abiotic (e.g., climate change) problems based on artificial and natural selections.Gene diversity can be a reflector of the genetic quality of seed crop, as well as its commercial value and choosing by the forest owner.
Group coancestry represents the probability that two genes in a population are IBD.Diversity refers to differences, and gene diversity indicates differences in genes.Notably, 1-group coancestry (Θ) denotes the probability that the genes are non-identical and thus diverse (i.e., gene diversity).
The number of "ideal" trees shows the trees that have the same gene diversity as the considered population.The status number (N s ) is a way of expressing group coancestry as an effective number.An appealing property of the status number is that, for a population of unrelated, non-inbred individuals, it is equal to the census number.The status number connects to the familiar concept of population size.
The status number has similarities with "effective number" in the classical sense, as it predicts inbreeding using random mating.Status number (N s ) is equal to half the inverse of group coancestry.It is also related to gene diversity [5,10,32,64]: When clones are unrelated and non-inbred, the status numbers of female (N s(f ) ) and male parents (N s(m) ) are calculated as [10,11,18,42]: where f i and m i correspond to the fertility of females and males of clone i, and N is the census number in the seed orchard.Fertility is estimated based on the strobilus assessment.
Status number (N s ) based on total fertility (i.e., clone fertility) is calculated as follows [11,16,35,39]: where ψ f and ψ m are the fertility variations in female and male parents, equivalent to CV f 2 + 1 and CV m 2 + 1, respectively.r is the correlation coefficient between female and male fertility.
The effective number of parents (N p ) can be defined as the number of genotypes divided by the sibling coefficient (Ψ) [15].This is further divided into the effective number of female parents [N p(f ) = N/(CV f 2 + 1)] and the effective number of male parents [N p(m) = N/(CV m 2 + 1)].Considering the correlations between female and male fertility, N p is calculated as follows [16,35]: Here, CV f and CV m are the coefficients of variation in female and male fertility, respectively, r is the correlation coefficient between female and male fertility, and N is the number of individuals.Fertility is estimated based on the flowering assessment [66].

Covariance between Female and Male Fertility
Under various scenarios of female and male fertility covariations (correlation), the effective number of parents is stochastically simulated across a range of correlation coefficients [67] (Figure 3).Generally, when there is no or limited covariation in female and male parental reproductive output fertility, the effective number of parents (N p ) is equivalent to the census number (N), assuming the seed orchard parents are unrelated and non-inbred.Positive covariations in female and male parental reproductive output fertility increase the parental fertility variations (Ψ), as this is affected by variations in both females (ψ f ) and males (ψ m ), leading to a decline in the effective number of parents (Figure 3a-c).On the other hand, negative covariations in female and male parental reproductive output fertility mitigates the asymmetrical variations between ψ f and ψ m (fertility variation imbalances), increasing the effective number of parents (Figure 3d-f).

Sibling Coefficient and Relative Effective Number
The sibling coefficient (Ψ) can be interpreted as the likelihood of two random gametes being identical by descent in a set of gametes from the same group, considering fertility variations [2].Thus, Ψ = 1 means that there that individuals made an equal contribution to the gamete gene pool in the population.When an equal number of seeds is collected Forests 2023, 14, 2172 9 of 14 from each tree, the female fertility is constant; thus, CV f = 0 (i.e., equal contribution among seed parents).Under equal seed harvesting conditions, the effective number of parents (Equation ( 18)) can be simplified [33,35,37] as: The relative effective number of parent or relative status number (N r ) is the proportion of the status number (N s ) or effective number of parents (N p ) to the census number (N), as follows: The relative effective number of parents for female (N r(f) ) and male fertility (N r(m) ) are also estimated based on female (ψ f ) and male fertility (ψ m ), as follows: The relative effective number of parent (N r ) and relative status number (N r ) for the total gene pool is estimated based on total fertility variation (Ψ), as follows:

.5. Parental Balance Curve and Maleness Index
Parental balance curves are shown as an example in Figure 4 [17] using cumulative gamete contribution.Parental balance can be assessed using a cumulative gamete contribution curve [18, 23,25,37,39].This is an important guide tool for plant geneticists.

Sibling Coefficient and Relative Effective Number
The sibling coefficient () can be interpreted as the likelihood of two random gametes being identical by descent in a set of gametes from the same group, considering fertility variations [2].Thus,  = 1 means that there that individuals made an equal contribution to the gamete gene pool in the population.When an equal number of seeds is collected from each tree, the female fertility is constant; thus, CVf = 0 (i.e., equal contribution among seed parents).Under equal seed harvesting conditions, the effective number of parents (Equation ( 18)) can be simplified [33,35,37] as: The relative effective number of parent or relative status number (Nr) is the proportion of the status number (Ns) or effective number of parents (Np) to the census number (N), as follows: The relative effective number of parents for female (Nr(f)) and male fertility (Nr(m)) are also estimated based on female (ψf) and male fertility (ψm), as follows: The relative effective number of parent (Nr) and relative status number (Nr) for the total gene pool is estimated based on total fertility variation (Ψ), as follows:

Parental Balance Curve and Maleness Index
Parental balance curves are shown as an example in Figure 4 [17] using cumulative gamete contribution.Parental balance can be assessed using a cumulative gamete contribution curve [18, 23,25,37,39].This is an important guide tool for plant geneticists.Variations in the parental contribution among families could be described using the parental cumulative curve shown in Figure 4.The cumulative contribution curve is linear, and the dotted diagonal line in Figure 4 denotes equal fertility among trees.
Maleness index (M i ) was defined as the proportion of a clone's reproductive success that is transmitted through its pollen, that is, by paternal parents (Figure 5) [31,32,68].The Mi represents the sexual asymmetry of parental contribution to seed crops among clones and provides a quantitative measure of gender [68][69][70] under some assumptions, such as Forests 2023, 14, 2172 10 of 14 equal fertility, and equality between ovule and pollen production.The M i based on female and male fertility (e.g., female and male strobilus production) was calculated as follows: where m i is the proportion of male strobilus production and f i is that of female strobilus production of the ith clone.Femaleness index equals 1 − M i , denoting the proportion of reproductive success transmitted by maternal parents.The high maleness of a clone indicates that the clone is contributing more as a father than as a mother parent when compared with other clones in the orchard [31,70].
Maleness index (Mi) was defined as the proportion of a clone's reproductive s that is transmi ed through its pollen, that is, by paternal parents (Figure 5) [31,32,68 Mi represents the sexual asymmetry of parental contribution to seed crops among and provides a quantitative measure of gender [68][69][70] under some assumptions, s equal fertility, and equality between ovule and pollen production.The Mi based on and male fertility (e.g., female and male strobilus production) was calculated as fol

𝑀 =
where mi is the proportion of male strobilus production and fi is that of female str production of the ith clone.Femaleness index equals 1 − Mi, denoting the propor reproductive success transmi ed by maternal parents.The high maleness of a clon cates that the clone is contributing more as a father than as a mother parent when pared with other clones in the orchard [31,70].The synchronization of flowering plays a vital role in assessing fertility vari parental balance curves and the determination of the maleness index.Floral synchr closely tied to phenology, which is the most significant factor affecting the mating s and flower pollination in seed orchards [71,72].For instance, in a second-generation seed orchard of the Chinese fir, Cunninghamia lanceolata, a positive correlation w served between the phenological synchronization index and both seed and cone prod [73].Moreover, by utilizing the phenological overlap index (proposed by [74]), resea calculated the level of synchronicity required for optimal seed yield in a clonal seed o of northern red oak (Quercus rubra) and found that a high index was essential [75].
Furthermore, various reproductive indicators, such as cone length, cone weig tile scale count, a proportion of aborted ovules, a presence of empty and filled seed ferred to as seed efficiency), seed weight, ratio of empty to developed seeds, and w of filled seeds and cones, should all be considered [76].Additionally, when assessi fertility of seed orchard crops, it is crucial to determine the depth of genetic diver The synchronization of flowering plays a vital role in assessing fertility variations, parental balance curves and the determination of the maleness index.Floral synchrony is closely tied to phenology, which is the most significant factor affecting the mating system and flower pollination in seed orchards [71,72].For instance, in a second-generation clonal seed orchard of the Chinese fir, Cunninghamia lanceolata, a positive correlation was observed between the phenological synchronization index and both seed and cone production [73].Moreover, by utilizing the phenological overlap index (proposed by [74]), researchers calculated the level of synchronicity required for optimal seed yield in a clonal seed orchard of northern red oak (Quercus rubra) and found that a high index was essential [75].
Furthermore, various reproductive indicators, such as cone length, cone weight, fertile scale count, a proportion of aborted ovules, a presence of empty and filled seeds, (referred to as seed efficiency), seed weight, ratio of empty to developed seeds, and weight of filled seeds and cones, should all be considered [76].Additionally, when assessing the fertility of seed orchard crops, it is crucial to determine the depth of genetic diversity in future generations and their ability to withstand unpredictable environmental challenges.The range of variation in reproductive success can serve as a fundamental basis for this evaluation [77].The entire process of fertility variations can be comprehensively examined, extending from seed germination tests to growth tests.

Conclusions
This comprehensive review provides valuable insights into fertility variations and their linkage parameters in the field of forest population genetics.Understanding the differences in reproductive success among individuals, known as fertility variations, has significant implications for gene conservation, seed production programs, managing forest genetic resources, and evolutionary and physiological studies.
Through an extensive analysis of the existing literature and knowledge, we created a comprehensive guide for future research on fertility variations and gene diversity.This review fills a notable gap in the literature by addressing the lack of a dedicated review paper specifically focusing on fertility variations and their linkage parameters.Fertility variation estimations, particularly through reproductive character assessments like cone production, have emerged as a cost-effective and widely used tool in plant sciences.They have found applications in seed orchards, seed stands, plantations, and natural populations.The increasing importance and popularity of fertility variations can be attributed to their numerous advantages and the ongoing research advancements.
We explored various methodologies for estimating fertility variations, including variations in female and male fertilities, total fertility variation (i.e., sibling coefficient), and effective parent numbers.Additionally, we discussed important linkage parameters such as coancestry, group coancestry, gene diversity, status number, and the effective number of parents.These parameters provide valuable insights into relatedness and gene diversity within forest populations, aiding in the management and conservation of forest genetic resources.
Furthermore, we highlighted the significance of parental balance curves as a tool for evaluating the contribution of individual parents to the overall gene pool.These curves visually represent cumulative gamete contribution and serve as a guide for breeding programs and seed production efforts.To further advance the field, we recommend incorporating molecular studies to support fertility research (i.e., marker-assisted selection (MAS)).Developing specialized stochastic software that is specifically designed for fertility variation and linkage parameter estimations would streamline calculations and analyses, leading to enhanced accuracy and efficiency.Moreover, fertility variations and their linkage parameters can be applied to new purposes, such as the establishment and selection of seed sources to produce climate-resilient seed crops based on gene diversity.
In summary, this review paper fills an important gap in the existing literature by providing a comprehensive overview of fertility variations and their linkage parameters.It serves as an invaluable resource for researchers and practitioners interested in studying and applying fertility variations in forest sciences.By understanding fertility variations, we can significantly contribute to gene conservation efforts, improve seed production programs, and gain deeper insights into the evolutionary and physiological aspects of forest populations.The integration of molecular studies, the development of specialized software, and the expansion of applications in seed source establishment and selection for climate resilience further advance the field of fertility research.

Figure 1 .
Figure 1.Scenarios of the parental contribution in model seed orchards with five unrelated, noninbred individuals (N = 5).The different part in the pie chart is the different contribution of each individual.Reproduced with permission from [11].

Figure 1 .
Figure 1.Scenarios of the parental contribution in model seed orchards with five unrelated, noninbred individuals (N = 5).The different part in the pie chart is the different contribution of each individual.Reproduced with permission from [11].

Forests 1 Figure 3 .Figure 3 .
Figure 3. Stochastic simulation of the effective number of parents (Np) with female and male fertility variations (CVf, CVm) under various covariation (correlation coefficients, r) between female and male reproductive outputs, where the census number is 100 (N = 100) in the population.Reproduced with permission from [67].

Figure 4 .Figure 4 .
Figure 4. Parental balance curves for reproductive outputs in a population of Taurus cedar.Reproduced with permission from [17].

Figure 5 .
Figure 5. Maleness index calculated for individual clones of 99) based on the observations of lus production for 4 years.The horizontal line shows where the average contribution for rep tion is the same for both genders.Reproduced with permission from[68].

Figure 5 .
Figure 5. Maleness index calculated for individual clones of 99) based on the observations of strobilus production for 4 years.The horizontal line shows where the average contribution for reproduction is the same for both genders.Reproduced with permission from [68].