How many pairwise copulas in a $k$-dimensional R-vine?

Question

I am studying R-vine copulas. Let $k$ denote the number of random variables the joint distribution of which we are modeling. R-vine breaks down the $k$-variate copula into $K$ bivariate copulas. I am trying to understand how $K$ is related to $k$.

Question: How many pairwise copulas are there in a $k$-dimensional R-vine?

Answer 1

My guess is, $K=k(k-1)/2$.

The tree structure in Figure 1 of Dissmann et al. (2013) illustrates that an R-vine is represented by $k-1$ trees that have $k-1,k-2,\dots,1$ edges, respectively, where each edge stands for a pair copula. This is an arithmetic progression the sum of which is: $(k-1)+(k-2)+\dots+1=k(k-1)/2$.
(My uncertainty is, is this the general case, or is it just a special case. It seems to me it is the general case.)

Reference

• Dissmann et al. "Selecting and estimating regular vine copulae and application to financial returns" (2013) (arXiv version here).