9 10 Apple (Malus spp.) is a widely grown and valuable fruit crop. Leaf shape and size are important 11 for flowering in apple and may also be early indicators for other agriculturally valuable traits. 12We examined 9,000 leaves from 869 unique apple accessions using linear measurements and 13 comprehensive morphometric techniques. We identified allometric variation in the length-to-14 width aspect ratio between accessions and species of apple. The allometric variation was due to 15 variation in the width of the leaf blade, not length. Aspect ratio was highly correlated with the 16 primary axis of morphometric variation (PC1) quantified using elliptical Fourier descriptors 17 (EFDs) and persistent homology (PH). While the primary source of variation was aspect ratio, 18 subsequent PCs corresponded to complex shape variation not captured by linear measurements. 19After linking the morphometric information with over 122,000 genome-wide SNPs, we found 20 high narrow-sense heritability values even at later PCs, indicating that comprehensive 21 morphometrics can capture complex, heritable phenotypes. Thus, techniques such as EFDs and 22 PH are capturing heritable biological variation that would be missed using linear measurements 23 2 alone, and which could potentially be used to select for a hidden phenotype only detectable using 24 comprehensive morphometrics. 25 26 Passiflora (8). Previous work used EFDs to assess apple fruit shape (9), but this technique has 47 not yet been applied to apple leaves. A newly developed morphometric technique, persistent 48 homology (PH), provides another method for estimating leaf shape. PH, like EFDs, is 49 normalized to differences in size, but it also could be orientation invariant. PH treats the pixels of 50 a contour as a 2D point cloud before applying a neighbor density estimator to each pixel. A 51 series of annulus kernels of increasing radii are used to isolate and smooth the contour densities. 52The number of connected components is recorded as a function of density for each annulus, 53 resulting in a curve (a reduced version of persistent barcode) that quantifies shape as topology. 54The topology-based PH approach can also be applied to serrations and root architecture, allowing 55

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