Starting from the root, extend k branches and append k children with probability p, or terminate with probability q=1–p. Then, we have a finite k-ary tree with probability one if 0 ≤p ≤1/k. Moreover, we give the expectation and variance of the length of ideal codewords for representing the finite trees. Furthermore, we establish the probability of obtaining infinite tree, that is, of penetrating to infinity without termination for case 1/k ≤p ≤1.