  We study the {\sc multicut on trees} and the {\sc generalized multiway Cut on
trees} problems. For the {\sc multicut on trees} problem, we present a
parameterized algorithm that runs in time $O^{*}(\rho^k)$, where $\rho =
\sqrt{\sqrt{2} + 1} \approx 1.555$ is the positive root of the polynomial
$x^4-2x^2-1$. This improves the current-best algorithm of Chen et al. that runs
in time $O^{*}(1.619^k)$. For the {\sc generalized multiway cut on trees}
problem, we show that this problem is solvable in polynomial time if the number
of terminal sets is fixed; this answers an open question posed in a recent
paper by Liu and Zhang. By reducing the {\sc generalized multiway cut on trees}
problem to the {\sc multicut on trees} problem, our results give a
parameterized algorithm that solves the {\sc generalized multiway cut on trees}
problem in time $O^{*}(\rho^k)$, where $\rho = \sqrt{\sqrt{2} + 1} \approx
1.555$ time.
