The asymptotic value of the zeroth-order Randić index and sum-connectivity index for trees

The zeroth-order Randić index and sum-connectivity index are two indices based on the vertex degrees. They appeared in the topological formula for the total π-electron energy of conjugated molecules and attracted a lot of attention in recent years. Let Tn be the set of trees of order n. Suppose each tree in Tn is equally likely. We get that for almost every tree, the zeroth-order Randić index is among (r1 ± ε)n and the sum-connectivity index is among (r2 ± ε)n, where r1, r2 are some constants and ε is any positive real number.