A Prufer code of a labeled free tree with n nodes is a sequence of length n-2 constructed by the following sequential process: for i ranging from 1 to n-2 insert the label of the neighbor of the smallest remaining leaf into the ith position of the sequence, and then delete the leaf. Prufer codes provide an alternative to the usual representation of trees. We present an optimal O(logn) time, n/logn processor EREW-PRAM algorithm for determining the Prufer code of an n-node labeled chain and an O(logn) time, n processor EREW-PRAM algorithm for constructing the Prufer code of an n-node labeled free tree. This resolves an open question posed by Wang et al. (IEEE Trans. Parallel Distributed Systems 8 (12) (1997) 1236-1240).