We study the statistics of growing networks with a tree topology in which each link carries a weight (k(i) k(j))(theta) , where k(i) and k(j) are the node degrees at the end points of link ij . Network growth is governed by preferential attachment in which a newly added node attaches to a node of degree k with rate A(k) =k+lambda . For general values of theta and lambda , we compute the total weight of a network as a function of the number of nodes N and the distribution of link weights. Generically, the total weight grows as N for lambda>theta-1 and superlinearly otherwise. The link weight distribution is predicted to have a power-law form that is modified by a logarithmic correction for the case lambda=0 . We also determine the node strength, defined as the sum of the weights of the links that attach to the node, as function of k . Using known results for degree correlations, we deduce the scaling of the node strength on k and N .