Asymptotic properties of the inversion number in colored trees

We consider a $b$-ary plane rooted tree $T$ whose vertices are colored independently and equiprobably in $m$ colors labelled with letters of an alphabet $\mathcal{A}=\left\{ A_{1}<A_{2}<...<A_{m}\right\} .$ A vertex $u\in T$ is an ancestor of a vertex $v\in T$ ($u\prec v),$ if the path leading along the edges from the root of the tree to the vertex $v$ passes through the vertex $u$. Denote $\text{col}(u)$ the color of the vertex $u.$ The coloring of the pair $u\prec v$ forms an inversion if $\text{col}(u)>\text{col}(v).$ We study the probabilistic characteristics of the total number of inversions in a colored $b$-ary plane rooted tree of a fixed height and the distribution of random variables that are functionals of the number of inversions in the subtrees of such a tree.