Before revision:  It is well known that this point location structure has {\cal O}(n) expected size and {\cal O} (n \ln n) expected construction time. Our main result is an improved tail bound, with exponential decay, for the size of the TSD: There is a constant such that the probability for a TSD to exceed its expected size by more than this factor is at most 1/e^n.This yields improved bounds on the TSD construction and their maintenance.I.e. TSD construction takes with high probability {\cal O}( n \ln n) time and TSD size can be made worst case {\cal O}(n) with an expected rebuild cost of {\cal O}( 1).
After revision:  It is well known that this point location structure has {\cal O}(n) expected size and {\cal O} (\log n) comparisons in the first and w.h.p. {\cal O}( n \ln n) time and TSD size can be made worst case {\cal O}(n) with an expected rebuild cost of {\cal O}( 1).