# 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.