Eye-Pleasing City Park Tour

There is a city park represented as a tree with $$$n$$$ attractions as its vertices and $$$n - 1$$$ rails as its edges. The $$$i$$$-th attraction has happiness value $$$a_i$$$.

Each rail has a color. It is either black if $$$t_i = 0$$$, or white if $$$t_i = 1$$$. Black trains only operate on a black rail track, and white trains only operate on a white rail track. If you are previously on a black train and want to ride a white train, or you are previously on a white train and want to ride a black train, you need to use $$$1$$$ ticket.

The path of a tour must be a simple path — it must not visit an attraction more than once. You do not need a ticket the first time you board a train. You only have $$$k$$$ tickets, meaning you can only switch train types at most $$$k$$$ times. In particular, you do not need a ticket to go through a path consisting of one rail color.

Define $$$f(u, v)$$$ as the sum of happiness values of the attractions in the tour $$$(u, v)$$$, which is a simple path that starts at the $$$u$$$-th attraction and ends at the $$$v$$$-th attraction. Find the sum of $$$f(u,v)$$$ for all valid tours $$$(u, v)$$$ ($$$1 \leq u \leq v \leq n$$$) that does not need more than $$$k$$$ tickets, modulo $$$10^9 + 7$$$.