In the MULTICUT problem, given an undirected graph G, a set of pairs of vertices P, and a budget k, the goal is to determine if there is a set S of at most k edges such that for each (s,t)∈P, the graph G−S has no path from s to t. In this article we first study the parameterized complexity of a variant of this problem, where the input graph is edge-weighted with arbitrary weights and the goal is to find a solution of minimum weight. Since weights are arbitrarily large, the weight of the solution is not a good choice for a parameter. The weighted problem is non-trivial even on trees and we study this problem on trees parameterized by structural parameters like the number of leaves and the request degree of every vertex. The studied parameters naturally interpolate the known polynomial time and NP-hardness results for this problem. We also give an FPT algorithm for another variant called WEIGHTED MULTICUT, where given an edge-weighted tree, the goal is to find a solution of size at most k edges that minimizes the weight.