Reference for Derivation of Higher order Runge Kutta I have a problem about determining $a_1, a_2, k_1, k_2, \ldots a_n, k_n,\ldots$
In the general form of the Higher Order Runge-Kutta below :
$$y_{r+1}=y_r+a_1k_1+a_2k_2+\cdots+a_nk_n$$
For the convenient, i'll write it down The Runge Kutta $2^{\text{nd}}$ in my book just in case if the formula is different from your views:
$$
\begin{align}
k_1&=hf(x_r,y_r)\\
k_2&=hf(x_r+p_1h,y_r+q_{11}k_1)\\ 
y_{r+1}&=y_r+(a_1k_1+a_2k_2)
\end{align}
$$
I'm not really sure, but some sources talk about Butcher Tableau, slope for each $k_n$, Taylor expansion, and rooted trees, for determining the $a_n$. And i don't really understand what their relation is, especially for Butcher Tableau, and rooted trees.
Could you explain all of these for me? Or just give me a good reference that discusses about derivation of the higher order Runge-Kutta method in detail, please?
Because, some books just skip over the derivation part.

A good source on Butcher tableaus and B-trees would, almost canonically, be the book by Butcher himself. For a short overview, see the three sets of slides with an introduction to B-trees, demonstration of the method for up to order 4 and outlook to implicit methods, or a historical overview
You could of course also read the most original source, the introduction of Runge-Kutta methods by M. Wilhelm Kutta (1901), "Beitrag zur näherungsweisen Lösung totaler Differentialgleichungen", where the order conditions are established and solved up to order 4, with an order 5 method that turned out not to be correct for systems.
