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Supporting Information Fig. S1 : Aerial photograph of the parental stand and of neighboring stands with indications on the dominant species composition.
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The white square corresponds to the parental stand and the yellow rectangles to recently naturally regenerated stands.

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Supporting Information Appendix S1: Validation of SSR genotypes through maternity and paternity analyses
Four of the 51 mother trees were not represented by grafts in the nursery but their genotype could be unambiguously reconstructed from the genotypes of their offspring. For the remaining progenies, we checked if the genotype of each offspring matched with the corresponding mother genotype. If a mismatch was found at more than one locus (out of 12), the offspring was excluded from the analysis. If only one mismatch was found, we re-genotyped and read again the SSR profiles of the mother/offspring couple. If these new analyses validated the mismatch, we excluded the offspring from the final dataset.
We then used CERVUS (Marshall et al. 1998) 1 to carry out a first paternity analysis. When the most likely father identified had a high SSR LOD score (>8) and exactly one mismatch with its offspring, we genotyped again the father/offspring pair to correct for any possible genotyping errors. We then used the subset of offspring genotyped at SNPs in a second paternity analysis with CERVUS . We found
144 trios (father/mother/offspring) with a high SNP LOD score (>40). If an inferred paternity relationship had a poor SSR LOD score (<8) and one or more mismatches with a given offspring but had a high SNP LOD score, we re-genotyped the corresponding father/offspring pair using SSRs. In this way, the SSR genotype of 90 fathers out of 260 (i.e. most of those with high reproductive success) could be controlled and corrected if needed.
In principle, discarding potential parental trees on the basis of a single mismatch could prove unjustified, as mutations can occur at SSRs. Actually, by comparing with the paternity analysis based on SNPs, we identified one father-tree with a high SNP LOD score that had two offspring presenting the same mismatch at one SSR locus. We interpreted this as a result of a mutation in the crown of the adult paternal tree; for subsequent analyses, we assigned to this tree the genotype matching with the two offspring. In any case, discarding completely a few offspring on the basis of a single mismatch with its mother is a conservative procedure that might eliminate a few legitimate offspring as well as many illegitimate ones but that should not create biases.
One possible concern with our approach is that, while all offspring benefitted from a quality control on the basis of allelic transmission from their mother, only part of them could benefit from a similar quality control from the father’s side, as the father of only a fraction of the offspring (50-60%) could be identified.
Marshall T.C., Slate J., Kruuk L.E.B. & Pemberton J.M. (1998). Statistical confidence for likelihood based paternity inference in natural populations. Mol. Ecol., 7, 639-655.

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However, we decided to correct all possible genotypes as we were interested in improving the estimates of model parameters, which depend on the likelihood of the inferred parentages
(following prescription of Oddou-Muratorio et al. 2003). Since the model used relies on fractional paternity analyses, true fathers having remaining mismatches with a given offspring (caused by genotyping errors or by a mutation) would still play a role in parameter estimates.
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Oddou-Muratorio S., Houot M.L., Demesure-Musch B. & Austerlitz F. (2003). Pollen flow in the wildservice tree,
Sorbus torminalis (L.) Crantz. I. Evaluating the paternity analysis procedure in continuous populations. Mol.
Ecol., 12, 3427-3439.

64 Supporting Information Fig. S2: The different spatial distributions used in simulations.
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Quercus robur trees are represented by grey diamonds, Q. petraea trees by black squares and intermediate trees by white triangles (genetic species assignment). A: Actual spatial distribution of the trees. B: One of the
100 input files corresponding to random spatial distribution of species. C: Disjoint spatial distribution of species in the stand.
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Supporting Information Fig. S3: Comparison of observed and predicted numbers of immigrant and hybrid offspring
A: Numbers of offspring of each mother tree resulting from Q. petraea pollen coming from inside or outside the stand (red and orange bars, respectively), of Q. robur pollen coming from inside or outside the stand (dark and light blue bars, respectively), and of pollen from admixed trees coming from inside the stand (green). 1)
Observed results. 2) Predicted results.
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B: Comparison between observed and predicted numbers of offspring originating from immigration for each mother tree.
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86 Black circles represent the progeny of Q. petraea and grey circles those of Q. robur.
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Supporting Information Appendix S2: Checking that observed hybridization rate is stronger with immigrant pollen than with local pollen
While fathers located inside the stand had been genotyped with a larger number of markers (12 SSRs
+ 262 SNPs) and could be directly assigned to species, the paternal species assignment of offspring produced by unknown fathers is only based on the likelihood for the father to belong to one species knowing the genotype of the offspring, the genotype of the mother and the allelic frequencies of the two species. In particular, some of the hybrids identified with this method might have admixed fathers. It was thus necessary to check that the difference in hybridization rates between local crosses and crosses involving immigrant pollen is not an artifact of the difference in the resolution of the methods of species assignment. For this purpose, we checked species assignment of all fathers using the same method (i.e. we used the indirect approach to assign the species of the father, even when the father had been identified through paternity analysis ) . We then used a χ² test to compare the observed and expected number of hybrids as a function of the type of cross for the two species
(Table 1). By comparing χ² robur
and χ² petraea
to χ² theoretical
(ddl=1) we could conclude that there was significantly more hybrids from immigrant than from local pollen for both species.
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Table 1: χ² test of the hypothesis that interspecific crosses are more frequent with immigrant pollen than with local pollen
Q. robur ♀
Crosses with local pollen
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Crosses with immigrant pollen
χ² robur
= 4.9 (p<0.05).
Interspecific crosses Intraspecific crosses
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35 850
Q. petraea ♀
Crosses with local pollen
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Crosses with immigrant pollen
χ² petraea
= 9.5 (p<0.005).
Interspecific crosses Intraspecific crosses
4 633
13 414
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These results cannot be influenced by paternal assignment quality. They support the idea that hybrid offspring are more likely to have been sired by immigrant pollen than by locally produced pollen.