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 Cn,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 Cn,k form a partition of the n-th Catalan numer Cn, that means Cn,1 + Cn,2 + · · ·+ Cn,n = Cn.