A partition of the Catalan numbers and enumeration of genealogical trees

A special relational structure, called genealogical tree, is introduced; its social interpretation and geometrical realizations are discussed. The numbers $C_{n,k}$ of all abstract genealogical trees with exactly n+1 nodes and k leaves is found by means of enumeration of code words. For each n, the $C_{n,k}$ form a partition of the n-th Catalan numer Cₙ, that means $C_{n,1}+C_{n,2}+ ...+C_{n,n} = Cₙ$.