Two algorithms for inserting a random element into a random heap are shown to be optimal (in the sense that they use the least number of comparisons on the average among all comparison-based algorithms) for di#erent values of n under a uniform model. Key words. Heaps, Hu#man coding scheme, expectedly optimal algorithms. The author is with the Institute of Statistical Science, Academia Sinica, 11529 Taipei, Taiwan. e-mail address: hkhwang@stat.sinica.edu.tw. 