Isometric relationship between the proportionality coefficients of the Montgomery and Montgomery–Koyama–Smith equations: Evidence from nine Malus cultivars
Danyu Zhou, Karl J. Niklas, Wangxiang Zhang, Wen Gu, Feixue Jiang, Peijian Shi
{"title":"Isometric relationship between the proportionality coefficients of the Montgomery and Montgomery–Koyama–Smith equations: Evidence from nine Malus cultivars","authors":"Danyu Zhou, Karl J. Niklas, Wangxiang Zhang, Wen Gu, Feixue Jiang, Peijian Shi","doi":"10.1007/s00468-026-02830-2","DOIUrl":null,"url":null,"abstract":"<div><h3>\n <b><i>Key message</i></b>\n </h3><p><b>There is a robust linear relationship between the proportionality coefficients of the Montgomery and Montgomery–Koyama–Smith equations.</b></p><h3>Abstract</h3><p>The Montgomery equation (ME) assumes a proportional relationship between individual leaf area and the product of leaf length and width, whereas the Montgomery<b>–</b>Koyama<b>–</b>Smith equation (MKSE) assumes a proportional relationship between the total leaf area per shoot and the product of the sum of individual leaf widths and maximum leaf length. Given the developmental homology between petals and leaves, the ME and MKSE are hypothesized to also apply to petal and corolla areas. However, prior studies have neglected the potential link between their respective proportionality coefficients. To validate the ME and MKSE for calculating individual petal area (<i>A</i>) and total corolla area per flower (<i>A</i><sub>T</sub>), and to test the correlation between the ME and MKSE proportionality coefficients (<i>k</i> for ME and <i>k</i><sub>KS</sub> for MKSE), we measured 2700 petals from 540 flowers across nine <i>Malus</i> cultivars. Goodness of fit was assessed using the root-mean-square error (RMSE) and the coefficient of determination (<i>r</i><sup>2</sup>). The ME and MKSE were validated for all nine cultivars, effectively predicting <i>A</i> and <i>A</i><sub>T</sub> with high precision (as gauged by low RMSE and high <i>r</i><sup>2</sup>). Importantly, a statistically significant one-to-one (isometric) scaling relationship was observed between <i>k</i> and <i>k</i><sub>KS</sub>. The isometric relationship between the two proportionality coefficients provides a quantitative link between individual petal geometry and whole-corolla structure, significantly simplifying the non-destructive estimation of corolla area (and potentially shoot-level total leaf area).</p></div>","PeriodicalId":805,"journal":{"name":"Trees","volume":"40 5","pages":""},"PeriodicalIF":2.3000,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Trees","FirstCategoryId":"2","ListUrlMain":"https://link.springer.com/article/10.1007/s00468-026-02830-2","RegionNum":3,"RegionCategory":"农林科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"FORESTRY","Score":null,"Total":0}
引用次数: 0
Key message
There is a robust linear relationship between the proportionality coefficients of the Montgomery and Montgomery–Koyama–Smith equations.
Abstract
The Montgomery equation (ME) assumes a proportional relationship between individual leaf area and the product of leaf length and width, whereas the Montgomery–Koyama–Smith equation (MKSE) assumes a proportional relationship between the total leaf area per shoot and the product of the sum of individual leaf widths and maximum leaf length. Given the developmental homology between petals and leaves, the ME and MKSE are hypothesized to also apply to petal and corolla areas. However, prior studies have neglected the potential link between their respective proportionality coefficients. To validate the ME and MKSE for calculating individual petal area (A) and total corolla area per flower (AT), and to test the correlation between the ME and MKSE proportionality coefficients (k for ME and kKS for MKSE), we measured 2700 petals from 540 flowers across nine Malus cultivars. Goodness of fit was assessed using the root-mean-square error (RMSE) and the coefficient of determination (r2). The ME and MKSE were validated for all nine cultivars, effectively predicting A and AT with high precision (as gauged by low RMSE and high r2). Importantly, a statistically significant one-to-one (isometric) scaling relationship was observed between k and kKS. The isometric relationship between the two proportionality coefficients provides a quantitative link between individual petal geometry and whole-corolla structure, significantly simplifying the non-destructive estimation of corolla area (and potentially shoot-level total leaf area).
期刊介绍：
Trees - Structure and Function publishes original articles on the physiology, biochemistry, functional anatomy, structure and ecology of trees and other woody plants. Also presented are articles concerned with pathology and technological problems, when they contribute to the basic understanding of structure and function of trees. In addition to original articles and short communications, the journal publishes reviews on selected topics concerning the structure and function of trees.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
