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 ${\displaystyle C_{n,k}}$ 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 ${\displaystyle C_{n,k}}$ form a partition of the n-th Catalan numer Cₙ, that means ${\displaystyle C_{n,1}+C_{n,2}+...+C_{n,n}=Cₙ}$.