The largest minimal rectilinear steiner trees for a set of n points enclosed in a rectangle with given perimeter

Suppose P is a set of points in the plane with rectilinear distance. Let l s (P) denote the length of a Steiner minimal tree for P. Let l r (P) denote the semiperimeter of the smallest rectangle with vertical and horizontal lines which encloses P. It is well known that l s (P) ≥ l r (P) for ∣P∣ ≥ 3 where ∣P∣ denotes the cordinality of P. In designing placement algorithms for printed circuits, l r (P) has been used as an estimate of l s (P) when ∣P∣ is small. Therefore, it is of some interest to know the value of\documentclass{article}\pagestyle{empty}\begin{document}$$ \rho _n \equiv \mathop {Max}\limits_{\left| P \right| = n} \frac{\ell{_s (P)}}{\ell{_r (P)}}. $$\end{document}In this paper we show ρ n tends to (√n+1)/2 and we give the exact value of ρ n for n ≤ 10.