sentences:  As in "join", note that the top part has two nodes that bracket the bottom part.Thus, we can simply "split" on each of these two nodes to divide the path into three parts, then "concatenate" the two outer ones so we end up with two parts, the top and bottom, as desired.In order to bound the competitive ratio for tango trees, we must find a lower bound on the performance of the optimal offline tree that we use as a benchmark.