This paper introduces the blossomed and grafted blossomed trees (BT and GBT, respectively) which are two new types of rooted trees. The trees consist of a finite number of solid and hollow vertices that represent buds and blossoms, respectively. Then the relation between them and derivative operators in a differential equation is analyzed. These concepts not only demonstrate how natural phenomena can be inspiring in mathematics, but also we can devise a method based on the BT and GBT for finding s-stage Runge–Kutta coefficients, with the appropriate degree of accuracy. However, solving higher-order differential equations in the general form entails dealing with numerous complex expressions, while the new algorithm based on the BT and GBT provides simplicity and practicability. One of the advantages of using BT and GBT is easier ordering and standardizing the relations derived from derivative operators in differential equations and synchronizing them with numerical methods such as the Runge–Kutta algorithms. In brief, this approach helps avoiding mistakes in spite of a high volume of processing operations. Moreover, the kth-order derivative for monotonically labeled GBT having n buds and blossoms using these types of trees with \(k+n\) buds and blossoms is studied. This strategy is also adopted for GBT without labeling.

本文介绍了开花和嫁接的开花树（分别为BT和GBT），这是两种新型的生根树。树木由有限数量的实心和空心顶点组成，分别代表芽和花朵。然后分析了它们与微分方程中微分算子之间的关系。这些概念不仅说明了自然现象如何在数学中得到启发，而且我们可以设计一种基于BT和GBT的方法，以适当的准确度来找到s级Runge-Kutta系数。然而，解决一般形式的高阶微分方程需要处理众多复杂的表达式，而基于BT和GBT的新算法则提供了简单性和实用性。使用BT和GBT的优点之一是易于对微分方程中的导数算子导出的关系进行排序和标准化，并使它们与数值方法（如Runge-Kutta算法）同步。简而言之，尽管处理操作量很大，但这种方法仍有助于避免错误。而且，研究了使用这些具有\（k + n \）芽和花的树的类型，对具有n个芽和花的单调标记GBT的k阶导数进行了研究。不带标签的GBT也采用此策略。