We review the existing and present new results on certain subtrees of the Galton-Watson family tree. For a positive integer N, define an N-ary subtree to be the tree of a deterministic N-splitting, rooted at the ancestor. Dekking (Amev. Math. Monthly 98:728–731, 1991) raised and answered the question how to compute the probability for a branching process to possess the binary splitting property, i.e., N = 2. Pakes and Dekking (J. Theor. Probab. 4:353–369, 1991) studied the general situation when N ≥ 2. Surprisingly, the case N ≥ 2 is studied so late, whereas the question for extinction of a branching process, i.e., non-existence of an infinite unary subtree (N = 1) has been studied extensively over the past 120–150 years.