Relation Between New Rooted Trees and Derivatives of Differential Equations

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.