Let*n*,*s*_{1},*s*_{2},...,*s*_{n} be non-negative integers and*M*(*s*_{1},*s*_{2},...,*s*_{n})=*a*_{1},*a*_{2},...,*a*_{n})|*a*_{i}is an integer and 0≤*a*_{i}≤*s*_{i} for each*i*∼. In this paper, the cardinality of maximum trees of finite sequences in*M*(*s*_{1},*s*_{2},...,*s*_{n}) is obtained, which generalizes some of Frankl’s results on families of finite sets with prescribed cardinalities for pairwise intersections.