  Let $G = (VG, AG)$ be a directed graph with a set $S \subseteq VG$ of
terminals and nonnegative integer arc capacities $c$. A feasible multiflow is a
nonnegative real function $F(P)$ of "flows" on paths $P$ connecting distinct
terminals such that the sum of flows through each arc $a$ does not exceed
$c(a)$. Given $\mu \colon S \times S \to \R_+$, the \emph{$\mu$-value} of $F$
is $\sum_P F(P) \mu(s_P, t_P)$, where $s_P$ and $t_P$ are the start and end
vertices of a path $P$, respectively.
  Using a sophisticated topological approach, Hirai and Koichi showed that the
maximum $\mu$-value multiflow problem has an integer optimal solution when
$\mu$ is the distance generated by subtrees of a weighted directed tree and
$(G,S,c)$ satisfies certain Eulerian conditions.
  We give a combinatorial proof of that result and devise a strongly polynomial
combinatorial algorithm.
