The Merrifield‐Simmons index σ( G) of a graph G is defined as the number of subsets of the vertex set, in which any two vertices are non-adjacent, i.e., the number of independent-vertex sets of G .B yT (n, k) we denote the set of trees with n vertices and with k pendent vertices. In this paper, we investigate the Merrifield‐Simmons index σ( T ) for a tree T in T (n, k) .F or all trees inT (n, k), we determined unique trees with the first and second largest Merrifield‐Simmons index, respectively. Let G = (V (G), E(G)) denote a graph whose set of vertices and set of edges are V (G) and E(G), respectively. For any v ∈ V (G), we denote the neighbors of v as NG(v) .B yn(G), we denote the number of vertices of G. All graphs considered here are both finite and simple.We denote, respectively, by Sn and Pn the star and path with n vertices. For any given graph G, its Merrifield‐Simmons index, simply denoted as σ( G), is defined as the number of subsets of the vertex set, in which any two vertices are non-adjacent, i.e., in graph-theoretical terminology, the number of independent-vertex subsets of G, including the empty set. For example, for the cycle C4 = v0v1v2v3, the independent-vertex subsets of V (C4) of all size are as follows: ∅, {v0}, {v1}, {v2}, {v3}, {v0 ,v 2}, {v1 ,v 3} ,a nd then σ( C4) = 7. As for the path Pn, σ( G) is exactly equal to the Fibonacci number Fn+2. This is perhaps why some researchers call the Merrifield‐Simmons index “Fibonacci number.” The concept of a (molecular) graph is introduced in [13], and discussed later in [1]. The Merrifield‐Simmons index for a molecular graph was extensively investigated in [10], where its chemical applications were demonstrated. In [6], Li et al. gave its other properties and applications. Wang and Hua [15] gave sharp lower and upper bounds for Merrifield‐Simmons index among all unicycle graphs.