For what it's worth, over the range N = [59050, 65536] = [3^10+1, 2^16], the number of choices in a ternary tree is 311 = 33, and in a base-2 tree is 216 = 32;

For what it's worth, over the range N = [59050, 65536] = [3^10+1, 2^16], the number of choices in a ternary tree is 311 = 33, and in a base-2 tree is 216 = 32; 65536 is the largest N for which binary wins over ternary. They tie a few times for even higher N, for the last time at the interval [3^17 + 1, 2^27] = [129140164, 134217728]; ternary trees with number of leaves in this interval have 18 levels, binary trees have 27. The point is that "sufficiently large" here turns out to be so large that nobody would actually build a phone tree that large, which is true for lots of mathematical results about "sufficiently large" numbers.