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Using the generating function formalism, we propose an exact formula for estimating betweenness centrality of an edge in finite tree-like components of random networks with arbitrary degree distributions. As a function of the degrees of two vertices interconnected by the edge and the size of the component to which the edge belongs, the formula can give an estimation of the mean value of the edge betweenness. We confirm the formula by simulations for Poisson and power law distributions respectively. We also study the asymptotic behavior of the formula and find that there is an asymptotic power law relation with an exponent 1.5 between the edge betweenness and the size of the component to which the edge belongs.