Abstract
Let L be a finite sequence of natural numbers. In Woryna (2017, 2018), we derived some interesting properties for the ratio ρn,L=|PRn(L)|∕|UDn(L)|, where UDn(L) denotes the set of all codes over an n-letter alphabet and with length distribution L, and PRn(L)⊆UDn(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 ρn,L is always greater than αn, where αn=(n−2)∕n for n>2 and α2=1∕6. Moreover, the number αn 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 αn. Namely, if L=(1,1,c), then ρn,L tends to (n−2)∕n with c→∞, and, if L=(1,2,c), then ρ2,L tends to 1∕6 with c→∞.
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