This is the zero skew tree problem where the tree is fixed.

Here we show how this problem can be solved in $O(n\log n)$ time by reducing it to minimum cost flow on 2-terminal series parallel graph. [1]

$C(v)$ denote the set of all the children of $v$.

For $vu\in E(T)$, and $u\in C(v)$, the graph $P(vu)$ is the parallel connection of one single edge with cost $c(vu)$, lower bound $\ell (vu)$, infinite upper bound and a series-parallel graph $S(u)$. Let the graph $S(v)$ for $v\in V(T)$ to be the series connection of $P(vu)$ for all $u\in C(v)$(the connection can be ordered arbitrarily).

The base case is when $u$ does not have any children, and $P(vu)$ is just a single edge with two terminals.

Let $G=S(r)$, where $r$ is the root of $T$. There is a bijection between the edges in $T$ and the edges in $G$.

Finding the minimum cost flow of value $t$ in $G$ with the two terminals gives us the desired solution by going back to the original edge using the bijection.

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