sentences:  Now, to join the bottom path to the top path, we simply "split" the top path between those two nodes, then "concatenate" the two resulting auxiliary trees on either side of the bottom path's auxiliary tree, and we have our final, joined auxiliary tree.Our "cut" operation will break a preferred path into two parts at a given node, a top part and a bottom part.More formally, it'll partition an auxiliary tree into two auxiliary trees, such that one contains all nodes at or above a certain depth in the reference tree, and the other contains all nodes below that depth.