Before revision:  The proposed analysis technique also shows that the expected depth is {\cal O}( \ln n ), which partially solves a recent conjecture by\emph{ [  Hemmer et al. that is used in the CGAL implementation of the TSD [ .
After revision:  The proposed analysis technique also shows that the expected depth is {\cal O}( \ln n ), which partially solves a recent conjecture by\emph{ [ and a conjecture on the depth of the DAG of Trapezoidal subdivisions Hemmer et al. that is used in the CGAL implementation of the TSD [ .