The total number of tree topologies for n tips for the rooted bifurcating case is known to be 1 x 2 x 3 x ... x (2n-3) which is (2n-3)! / (2^(n-2) (n-2)!), and the number of bifurcating LH is n! (n-1)! / 2^(n-1) The ratio of those is n! (n-1)! (n-2)!/(2n-3)! which for n=3 is 1 per topology, which is right as considering all cases shows. However, we would need to know that different tree topologies each have the same number of LH, and that isn't true as considering n=5 shows.
So, okay, I give up, but we at least can get a rough idea of the average number of LH per TT.