.Net R-Tree

"It needs to be sorted by FinishTime not starttime"

How so?  Sorting by end date you can miss an overlap.  Unless you scan the whole list which would defeat the purpose of a sort.

 // Sort by end date - Does not allow quit eairly.
1/1/09 - 1/3/09
               1/3/09 - 1/4/09 ** does not overlap
      1/2/09             -         1/5/09 ** overlaps 1, but would miss this.

"Sorted By Start Time

A  00:00->11:59

B  09:00->10:00

D  11:00->23:00

C  13:00->17:00

..

 Both sorting orders still misses the fact that

 A is overlapped by B & D"

Huh?  Sorted by start time does *not miss this.  9:00 is less than 11:59 so B returns true.   11:00 is less then 11:59 so D return true.  Testing C returns false as expected.  B and D overlap A.  Maybe not clear on your point.

base = A;

t(i+1).Start < base.End

All I said was sorting by start date works, and sorting by end date does not.  Given event X, find all other events that overlap works below.  Note: in this implementation, back-to-back meetings do not overlap, so I use "<" instead of "<=".

private static void TestOverlap()
{
    List<Event> list = new List<Event>();
    list.Add(new Event() { Start = "00:00".ToDate(), End = "11:59".ToDate(), Description = "A" });
    list.Add(new Event() { Start = "11:00".ToDate(), End = "23:00".ToDate(), Description = "D" });
    list.Add(new Event() { Start = "13:00".ToDate(), End = "17:00".ToDate(), Description = "C" });
    list.Add(new Event() { Start = "09:00".ToDate(), End = "10:00".ToDate(), Description = "B" });
    list.Add(new Event() { Start = "00:00".ToDate(), End="11:59:30".ToDate(), Description = "E"});
    Event source = list[0]; // Find events that overlap this event.

    // Sort collection by start date using default comparer on Event class.
    list.Sort();
    Console.WriteLine("Events sorted:");
    list.WriteEnumerable();

    // Get the new index of source in the sorted list.
    int index = list.BinarySearch(source);

    // Take overlapped events.
    var q = from ev in list.Skip(index + 1).TakeWhile(e => e.Start < source.End)
            select ev;
    Console.WriteLine("\nEvents that overlap event: {0}", list[0]);
    q.WriteEnumerable();
}