Rotation distance is a fundamental parameter in the study of tree balancing and data structures. One of the fundamental open problem in the area is to compute the minimum number of rotations required to transform one tree to the other, in which both trees have same number of nodes. We define shallow rotation distance with parameter k (which we call k-shallow rotation distance, denoted by \(d_k(T_1,T_2)\) ) to be the rotation distance between any two full binary trees when the rotation is allowed only at nodes within depth k. Indeed, 2-shallow rotation distance is well defined and is already the frontier of unknown territory in algorithmic approaches to rotation distance problem. In this paper, we identify a special case of rotation distance problem, known as the left- \(\lambda \) - \(\text {RotDist}\) and right- \(\lambda \) - \(\text {RotDist}\) and prove that they are as hard as the general version of the problem. Complementing this, we give a polynomial time algorithms for computing the \(d_2(T_1,T_2)\) in three scenarios  (a)  when \(T_1\) is a right- \(\lambda \) -tree and \(T_2\) is a right-comb tree,  (b)  when \(T_1\) is a left- \(\lambda \) tree and \(T_2\) is a left-comb-tree and  (c) when \(T_1\) is a left-right- \(\lambda \) -tree and \(T_2\) is a double comb with matching subtree size, where a double-comb is a tree in which all the nodes are present only on the right-most or left-most path from the root. We obtain our results by using a known connection between the rotation distance and the well-studied algebraic object called Thompson’s group \(\mathcal {F}\) [8]. We use the forest representation for a pair of binary trees introduced by Belk and Brown (2005), where we design a new measure for pairs of spaces in the representation, and demonstrate that this new measure indeed captures the shortest representation of the group element in the above three scenarios.