For a family of graphs (formula presented), the (formula presented) -Contraction problem takes as an input a graph G and an integer k, and the goal is to decide whether there exists (formula presented) of size at most k such that G/F belongs to (formula presented). When (formula presented) is the family of paths, trees or cacti, then the correspond-ing problems are Path Contraction, Tree Contraction and Cac-tus Contraction, respectively. It is known that Tree Contraction and Cactus Contraction do not admit a polynomial kernel unless (formula presented), while Path Contraction admits a kernel with O(k) vertices. The starting point of this article are the following natural ques-tions: What is the structure of the family of paths that allows Path Con-traction to admit a polynomial kernel? Apart from the size of the solu-tion, what other additional parameters should we consider so that we can design polynomial kernels for these basic contraction problems? With the goal of designing polynomial kernels, we consider the family of trees with bounded number of leaves (note that the family of paths are trees with at most two leaves). In particular, we study Bounded Tree Contraction (Bounded TC). Here, an input is a graph G, integers k and, (formula presented) and the goal is to decide whether or not, there exists a subset (formula presented) of size at most k such that G/F is a tree with at most leaves. We design a kernel for Bounded TC with O(k) vertices and (formula presented) edges. Finally, we study Bounded Cactus Contraction (Bounded CC) which takes as input a graph G and integers k and. The goal is to decide whether there exists a subset (formula presented) of size at most k such that G/F is a cactus graph with at most (formula presented) leaf blocks in the corresponding block decomposi-tion. For Bounded CC we design a kernel with (formula presented) vertices and (formula presented) edges. We complement our results by giving kernelization lower bounds for Bounded TC, Bounded OTC and Bounded CC by showing that unless (formula presented) the size of the kernel we obtain is optimal.