  For a family of graphs $\cal F$, the $\mathcal{F}$-Contraction problem takes
as an input a graph $G$ and an integer $k$, and the goal is to decide if there
exists $S \subseteq E(G)$ of size at most $k$ such that $G/S$ belongs to $\cal
F$. Here, $G/S$ is the graph obtained from $G$ by contracting all the edges in
$S$. Heggernes et al.~[Algorithmica (2014)] were the first to study edge
contraction problems in the realm of Parameterized Complexity. They studied
$\cal F$-Contraction when $\cal F$ is a simple family of graphs such as trees
and paths. In this paper, we study the $\mathcal{F}$-Contraction problem, where
$\cal F$ generalizes the family of trees. In particular, we define this
generalization in a "parameterized way". Let $\mathbb{T}_\ell$ be the family of
graphs such that each graph in $\mathbb{T}_\ell$ can be made into a tree by
deleting at most $\ell$ edges. Thus, the problem we study is
$\mathbb{T}_\ell$-Contraction. We design an FPT algorithm for
$\mathbb{T}_\ell$-Contraction running in time
$\mathcal{O}((2\sqrt(\ell))^{\mathcal{O}(k + \ell)} \cdot n^{\mathcal{O}(1)})$.
Furthermore, we show that the problem does not admit a polynomial kernel when
parameterized by $k$. Inspired by the negative result for the kernelization, we
design a lossy kernel for $\mathbb{T}_\ell$-Contraction of size $
\mathcal{O}([k(k + 2\ell)] ^{(\lceil {\frac{\alpha}{\alpha-1}\rceil + 1)}})$.
