The asymptotic values of the general Zagreb and Randić indices of trees with bounded maximum degree

Let T ∆ n denote the set of trees of order n, in which the degree of each vertex is bounded by some integer ∆. Suppose that every tree in T ∆ n is equally likely. We show that the number of vertices of degree j in T ∆ n is asymptotically normal with mean (μj + o(1))n and variance (σj + o(1))n, where μj , σj are some constants. As a consequence, we give estimate to the value of the general Zagreb index for almost all trees in T ∆ n . Moreover, we obtain that the number of edges of type (i, j) in T ∆ n also has mean (μij + o(1))n and variance (σij + o(1))n, where an edge of type (i, j) means that the edge has one end of degree i and the other of degree j, and μij , σij are some constants. Then, we give estimate to the value of the general Randić index for almost all trees in T ∆ n .