Trees are a common structure to represent the intertask communication pattern of a parallel algorithm. In this paper, we consider the embedding of a complete binary tree in a star graph with the objective of minimizing congestion and dilation. We develop two embeddings: (i) a congestion-free, dilation-2, load-1 embedding of a level-p binary tree, and (ii) a congestion-free, dilation-2, load-2(k) embedding of a level-(p + k) binary tree, into an n-dimensional star graph, where p = Sigma(i=2)(n) [log i] = Omega(n log n) and k is any positive integer. The first result offers a tree of size comparable or superior to existing results, but with less congestion and dilation. The second result provides more flexibility in the embeddable tree sizes compared to existing results. (C) 1999 John Wiley & Sons, Inc.