
Genetic diversity analysis of F1 progeny in heart-shaped walnuts
Using 14 SSR markers, genetic diversity analysis was conducted on the female parent and 32 F1 offspring of heart-shaped walnuts. A total of 87 alleles were obtained from the 14 SSR markers (Table 1). The number of alleles (Na) of SSR markers was 3 (Jr11) ~ 13 (Jr12), with an average of 6.214. The number of effective alleles (Ne) ranged from 1.921 (Jr55) to 7.930 (Jr12), with an average of 3.375. The major allele frequency (MAF) ranged from 0.242 (Jr12) to 0.667 (Jr55), with an average of 0.462. The frequency of invalid alleles was − 0.139 (Jr29) ~ 1.00 (Jr05, Jr44, Jr50, Jr52), with an average of 0.644. The observed heterozygosity (Ho), expected heterozygosity (He), and unbiased expected heterozygosity (uHe) were 0 ~ 1.00 (Jr29), 0.479 (Jr55) ~ 0.874 (Jr12), and 0.487 (Jr55) ~ 0.889 (Jr12), respectively. The genetic diversity (GD), Shannon index (I), and the polymorphism information content (PIC) were 0.479 (Jr55) ~ 0.888 (Jr12), 0.760 (Jr11) ~ 2.322 (Jr12), and 0.392 (Jr11) ~ 0.879 (Jr12), respectively. All 14 SSR markers had good polymorphism. Three SSR markers (Jr11, Jr44, and Jr55) showed moderate polymorphism, while 11 SSR markers showed high polymorphism. The fixation index was − 0.304 (Jr29) ~ 1.00 (Jr05, Jr11, Jr44, Jr50, Jr52, Jr55), with an average value of 0.669. Among them, the fixation index of Jr29 and Jr40 were less than 0, indicating the presence of heterozygous excess in the F1 offspring of heart-shaped walnuts.
Genetic diversity analysis of F1 offspring of Jinghong 1
The genetic diversity analysis of 12 SSR markers in Jinghong 1 and 45 F1 offspring showed that a total of 48 alleles were obtained from 12 SSR markers (Table 2). The Na values of SSR markers ranged from 2 (Jr40) to 9 (Jr44), with an average of 4.000. The Ne value was 1.240 (Jr40) ~ 4.474 (Jr44), with an average of 1.982. The MAF value ranged from 0.315 (Jr44) to 0.913 (Jr50), with an average of 0.707. The frequency of invalid alleles ranged from 0.004 (Jr29) to 0.999 (Jr15, Jr45), with an average of 0.704. The values of Ho, He, and uHe were 0.000 ~ 0.609 (Jr29), 0.162 (Jr50) ~ 0.776 (Jr44), and 0.163 (Jr50) ~ 0.785 (Jr44), respectively. The values of GD, I, and PIC were 0.162 (Jr50) ~ 0.776 (Jr44), 0.344 (Jr40, Jr50) ~ 1.744 (Jr44), and 0.154 (Jr50) ~ 0.745 (Jr44), respectively. Among them, the SSR markers with the highest values of Ho, He, uHe, GD, I, and PIC were Jr44, while the ones with the lowest values were Jr50. In the F1 offspring of Jinghong 1, three SSR markers (Jr40, Jr50, and Jr52) showed low polymorphism, six SSR markers (Jr12, Jr15, Jr35, Jr45, Jr53, and Jr56) showed moderate polymorphism, and three SSR markers (Jr29, Jr44, and Jr55) showed high polymorphism. The Fi value was 0.016 (Jr29) ~ 1.00, with the average value of 0.763. The average PIC value of 12 SSR markers in 45 F1 offspring of Jinghong 1 was 0.388, indicating the low probability of gene exchange events occurring in its offspring.
Identification of parent-child relationship in F1 offspring of heart-shaped walnuts
Parental exclusion probability and analysis of Hardy&Weinberg equilibrium were calculated using 14 SSR markers on 32 F1 offspring of heart-shaped walnuts (Table 3). When the paternal and maternal genotypes were unknown, the NE-1P value ranged from 0.394 to 0.883. When the genotype of the maternal parent was known, the NE-2P value ranged from 0.244 to 0.789. When the maternal parent was known, the marker with the lowest paternal exclusion rate was Jr12, and the marker with the highest exclusion rate was Jr11. When two parental genotypes were known, the NE-PP value ranged from 0.087 to 0.677. Under the conditions of unknown parental genotypes, known maternal genotypes, or known parental genotypes, the cumulative exclusion probability of 14 SSR markers was as high as 99%. In addition, NE-I and NE-SI values were 0.026 ~ 0.358 and 0.320 ~ 0.598, respectively. Among 14 SSR markers, 12 SSR markers had cumulative exclusion rates greater than 99% under known parental genotypes, namely Jr05, Jr11, Jr15, Jr29, Jr35, Jr44, Jr45, Jr50, Jr52, Jr53, Jr55, and Jr56. However, under the condition that the maternal genotype was known but the paternal genotype was unknown, only 9 SSR markers had cumulative exclusion rates greater than 99%, namely Jr05, Jr11, Jr29, Jr44, Jr45, Jr52, Jr53, Jr55, and Jr56. Therefore, these 9 SSR markers could serve as core SSR markers for the identification of hybrid offspring in heart-shaped walnuts. The results of the Hardy&Weinberg equilibrium test indicated that the corresponding loci of the four SSR markers (Jr40, Jr45, Jr53, and Jr55) conformed to the law of genetic balance. The corresponding loci of 5 SSR markers (Jr05, Jr35, Jr44, Jr52, and Jr56) significantly deviated from the law of genetic balance.
According to the genotyping results of the heart-shaped walnut and its F1 offspring (Table 4), Jr55 identified the highest number of maternal self-pollination offspring, with 21, accounting for 65.63%. The number of self-pollination offspring identified by Jr40 was the lowest, with 3, accounting for 9.38%. The SSR marker with the highest true offspring identification rate was Jr40, which was 87.50%. The SSR marker with the lowest probability was Jr50, which was 15.63%. On the contrary, the SSR marker with the highest probability of new genotypes appearing in offspring was Jr50, which was 84.38%. The SSR marker with the lowest probability was Jr40, which was 12.5%. This indicated that F1 offspring might have undergone a certain degree of gene exchange events. In addition, the genotype results of Jr29, Jr40, Jr45, Jr53, and Jr56 in F1 offspring indicated the presence of gene segregation. Further analysis of the F1 offspring containing the heart-shaped walnut maternal genotype revealed the presence of different paternal parents in the F1 offspring (Table 5). Jr29, Jr40, Jr45, Jr53, and Jr56 could identify 1, 5, 3, 2, and 1 paternal parent, respectively. The number of F1 offspring was 1, 25, 5, 5, and 3, respectively. The above results indicated that natural pollination enabled heart-shaped walnuts to receive pollen from different paternal parents, resulting in offspring with different genotypes.
Identification of parent-child relationship in F1 offspring of Jinghong 1
Parental exclusion probability and analysis of Hardy&Weinberg equilibrium were calculated using 12 SSR markers on 45 F1 offspring of Jinghong 1 (Table 6). When the paternal and maternal genotypes were unknown, the NE-1P value ranged from 0.599 to 0.987, with an average of 0.882. When the genotype of the maternal parent was known, the NE-2P value ranged from 0.421 to 0.920, with an average of 0.761. When the maternal parent was known, the marker with the lowest paternal exclusion rate was Jr44, and the marker with the highest exclusion rate was Jr50. When two parental genotypes were known, the NE-PP value ranged from 0.232 to 0.853, with an average of 0.634. When parental genotypes were both unknown, the cumulative exclusion rate of 12 SSR markers was only 79.8%. When the maternal parent was known, the cumulative exclusion rate of SSR markers was as high as 97%. When parental genotypes were both known, the cumulative exclusion rate of SSR markers was as high as 99%. In addition, NE-I and NE-SI values were 0.081 ~ 0.711 and 0.382 ~ 0.847, respectively, with an average of 0.402 and 0.639. Among the 12 SSR markers, 6 SSR markers had a cumulative exclusion rate of over 99% under known parental genotypes, namely Jr35, Jr40, Jr45, Jr50, Jr52, and Jr53. They could be used as SSR markers for identifying the offspring of Jinghong 1. The results of Hardy&Weinberg equilibrium test indicated that the corresponding locus of Jr29 conformed to the law of genetic balance, while the corresponding locus of Jr55 significantly deviated from the law of genetic balance.
According to the genotyping results of Jinghong 1 and its F1 offspring (Table 7), Jr50 identified the highest number of maternal self-pollination offspring, with 41, accounting for 91.11%. Jr35 identified the lowest number of maternal self-pollination offspring, which was 0. The SSR marker with the highest true offspring identification rate was Jr50, which was 91.11%. Next were Jr40 and Jr52, both accounting for 88.89%. The SSR marker with the lowest probability was Jr53, which was 15.56%. On the contrary, the SSR marker with the highest probability of new genotypes appearing in offspring was Jr53, accounting for 84.44%. The SSR marker with the lowest probability was Jr50, which was 8.89%. In addition, the genotype results of Jr29, Jr35, Jr44, Jr52, and Jr55 in the F1 offspring of Jinghong 1 indicated the presence of gene segregation. Further analysis of F1 offspring containing the genotype of Jinghong 1 revealed the presence of different paternal parents (Table 8). Jr29, Jr35, Jr44, Jr52, and Jr55 could identify 3, 2, 2, 1, and 1 paternal parent, respectively. The number of F1 offspring was 24, 15, 5, 1, and 1, respectively.
Analysis of the genetic relationship between heart-shaped walnut and its F1 offspring
The UPGMA clustering results based on genetic distance indicate (Table S1, Fig. 1) that the genetic distance between XX21 and the maternal parent was the closest, at 0.2561. The genetic distance between XX1 and the maternal parent was the farthest, at 0.7561. In F1 offspring, the genetic distance between XX6 and XX20 was the farthest (1.000), and the genetic distance between XX10 and XX12 was the closest (0.2857). At the genetic distance of 0.0864, 33 heart-shaped walnut samples were divided into two categories: group A and group B. Group A consisted of 6 F1 offspring, namely XX1, XX2, XX3, XX5, XX18, and XX19. In Group A, the genetic distance between XX2 and XX5 was relatively close, while the genetic distance between XX3 and XX19 was relatively close. At the genetic distance of 0.0851, Group B was further divided into two subclasses, B1 and B2. Group B1 contained three F1 offspring, namely XX13, XX14, and XX24. XX13 and XX14 first gathered together and then gathered with XX24 to form Group B1. At the genetic distance of 0.0305, group B2 could be further divided into two subclasses: B2-1 and B2-2. B2-1 contained 8 F1 offspring, namely XX4, XX10, XX12, XX16, XX17, XX20, XX28, and XX32. B2-2 contained the heart-shaped walnut maternal parent and 15 F1 offspring, namely XX6, XX7, XX8, XX9, XX11, XX15, XX21, XX22, XX23, XX25, XX26, XX27, XX29, XX30, and XX31. The female parent of heart-shaped walnuts was divided into the B2-2 group, indicating that the genetic distance between the 15 F1 offspring of the B2-2 group and the heart-shaped walnut was closer. The above results indicated that genetic variation had occurred in the F1 offspring of heart-shaped walnuts, and they had been influenced by natural pollination, which had to some extent changed the traits of the F1 offspring.
Cluster analysis between the female parent and 32 F1 offspring of heart-shaped walnuts.
Genetic structure analysis was conducted on the F1 offspring population of heart-shaped walnuts based on STRUCTURE (Table 9). When K = 3, ΔK had the maximum value, indicating that 32 F1 offspring were divided into 3 groups (Fig. 2). Group I contained 11 F1 offspring, namely XX8, XX9, XX10, XX11, XX12, XX16, XX17, XX20, XX25, XX28, and XX32. Group II contained 13 F1 offspring, namely XX6, XX7, XX13, XX14, XX15, XX21, XX22, XX23, XX26, XX27, XX29, XX30, and XX31. Group III contained 8 F1 offspring, namely XX1, XX2, XX3, XX4, XX5, XX18, XX19, and XX24. However, the Qi values of 7 F1 offspring were less than 0.8, which were XX8, XX16, XX25, XX32 in Group I, XX7, XX14, XX26 in Group II, and XX24 in Group III. Among them, the Qi values of 4 F1 offspring were less than 0.6, namely XX25 and XX32 in Group I, XX7 in Group II, and XX24 in Group III. The above explanation showed that the accuracy of the classification of F1 offspring of heart-shaped walnuts based on STRUCTURE was lower than that of cluster analysis, and further verification was needed.

Cluster analysis of the female parent and 45 F1 offspring of Jinghong 1.
Analysis of the genetic relationship between Jinghong 1 and its F1 offspring
The UPGMA clustering results based on genetic distance (Table S2, Fig. 2) indicated that the genetic distance between HR20 and Jinghong 1 was the closest, at 0.1911. The genetic distance between HR31 and HR44 and Jinghong 1 was the farthest, at 0.6911. Among 45 F1 offspring, HR05, HR12, and HR13 had the closest evolutionary relationship (0.0244), while HR09 and HR44 had the farthest evolutionary relationship (0.8577). At the genetic distance of 0.1978, 46 Jinghong 1 samples were divided into two groups: group A and group B. Group A contained 2 F1 offspring, namely HR8 and HR24. Group B contained 43 F1 offspring. At the genetic distance of 0.0372, group B was further divided into B1 and B2 groups. Group B1 contained 5 F1 offspring, namely HR1, HR9, HR25, HR31, and HR44. Among them, HR1 and HR9 first clustered together, HR31 and HR44 clustered together and then clustered with HR25 to form the B1 group. B2 contained 38 F1 offspring. At the genetic distance of 0.0381, group B2 was further divided into B2-1 and B2-2 groups. Among them, group B2-1 contained 34 F1 offspring, namely HR2, HR3, HR4, HR5, HR6, HR7, HR10, HR11, HR12, HR13, HR15, HR16, HR17, HR18, HR19, HR20, HR21, HR22, HR23, HR26, HR28, HR30, HR32, HR33, HR34, HR35, HR37, HR38, HR39, HR40, HR41, HR42, HR43, and HR45. In addition, the female parent of Jinghong 1 was divided into the B2-1 group, indicating the closer genetic relationship between these 34 F1 offspring. Among them, HR15, HR16, HR19, HR21, and HR28 were clustered on the same branch as the female parent of Jinghong 1, indicating that these 5 F1 progeny had a closer genetic relationship with the female parent and were more likely to be true progeny. The B2-2 group contained 4 F1 offspring, namely HR14, HR27, HR29, and HR36. HR27 and HR29 first clustered together, and then clustered together at HR36 and HR14 to form the B2-2 group.
Genetic structure analysis was conducted on the F1 offspring population of Jinghong 1 based on STRUCTURE (Table 10). When K = 3, ΔK had the maximum value, indicating that 45 F1 offspring were divided into 3 groups (Fig. 3). Group I contained 17 F1 offspring, namely HR6, HR7, HR8, HR10, HR14, HR16, HR22, HR24, HR27, HR30, HR32, HR35, HR36, HR38, HR40, HR43, and HR45. Group II contained 15 F1 offspring, namely HR5, HR12, HR13, HR15, HR17, HR18, HR19, HR20, HR21, HR26, HR28, HR34, HR37, HR41, and HR42. Group III contained 13 F1 offspring, namely HR1, HR2, HR3, HR4, HR9, HR11, HR23, HR25, HR29, HR31, HR33, HR39, and HR44. However, the Qi values of 14 F1 offspring were less than 0.8. Among them, the Qi values of 3 F1 offspring were less than 0.6, namely HR4, HR6, and HR7. This indicated that the classification problem of these three F1 offspring was worth further exploration.

Population genetic structure of F1 offspring in heart-shaped walnuts and Jinghong 1.
Screening specific walnut F1 progeny with excellent dry weight traits based on Jr35
Jr35 is significantly correlated with the dry weight of walnut nuts, and heterozygous walnuts have better nut dry weight traits than homozygous walnuts19. Among 32 F1 offspring of heart-shaped walnuts, XX1, XX2, XX3, XX5, XX18, and XX19 were heterozygous, while the remaining 26 offspring were homozygous. Therefore, it was speculated that the dry weight traits of nuts in XX1, XX2, XX3, XX5, XX18, and XX19 might be superior. Among 45 F1 offspring of Jinghong 1, HR6, HR7, HR10, HR15, HR16, HR17, HR18, HR19, HR22, HR23, HR28, HR29, HR36, HR43, and HR44 were heterozygous, while the remaining 30 offspring were homozygous. Therefore, it was speculated that the nut dry weight traits of HR6, HR7, HR10, HR15, HR16, HR17, HR18, HR19, HR22, HR23, HR28, HR29, HR36, HR43, and HR44 were more excellent.