On the proportion of prefix codes in the set of three-element codes

Let L be a finite sequence of natural numbers. In Woryna (2017, 2018), we derived some interesting properties for the ratio ρ_n,L=|PR_n(L)|∕|UD_n(L)|, where UD_n(L) denotes the set of all codes over an n-letter alphabet and with length distribution L, and PR_n(L)⊆UD_n(L) is the corresponding subset of prefix codes. In the present paper, we study the case when the length distributions are three-element sequences. We show in this case that the ratio ρ_n,L is always greater than α_n, where α_n=(n−2)∕n for n>2 and α₂=1∕6. Moreover, the number α_n is the best possible lower bound for this ratio, as the length distributions of the form L=(1,1,c) and L=(1,2,c) assure that the ratios asymptotically approach α_n. Namely, if L=(1,1,c), then ρ_n,L tends to (n−2)∕n with c→∞, and, if L=(1,2,c), then ρ_2,L tends to 1∕6 with c→∞.