A Practical Introduction to Data Structures and Algorithm Analysis

176 Chap. 5 Binary Trees37244273240 42235120Figure 5.15 An example of BST insertion. A record with value 35 is insertedinto the BST of Figure 5.13(a). The node with value 32 becomes the parent of thenew node containing 35.You should pay careful attention to the implementation for inserthelp.Note that inserthelp returns a pointer to a BSTNode. What is being returnedis a subtree identical to the old subtree, except that it has been modified to containthe new record being inserted. Each node along a path from the root to the parentof the new node added to the tree will have its appropriate child pointer assignedto it. Except for the last node in the path, none of these nodes will actually changetheir child’s pointer value. In that sense, many of the assignments seem redundant.However, the cost of these additional assignments is worth paying to keep the insertionprocess simple. The alternative is to check if a given assignment is necessary,which is probably more expensive than the assignment!The shape of a BST depends on the order in which elements are inserted. A newelement is added to the BST as a new leaf node, potentially increasing the depth ofthe tree. Figure 5.13 illustrates two BSTs for a collection of values. It is possiblefor the BST containing n nodes to be a chain of nodes with height n. This wouldhappen if, for example, all elements were inserted in sorted order. In general, it ispreferable for a BST to be as shallow as possible. This keeps the average cost of aBST operation low.Removing a node from a BST is a bit trickier than inserting a node, but it is notcomplicated if all of the possible cases are considered individually. Before tacklingthe general node removal process, let us first discuss how to remove from a givensubtree the node with the smallest key value. This routine will be used later by thegeneral node removal function. To remove the node with the minimum key valuefrom a subtree, first find that node by continuously moving down the left link untilthere is no further left link to follow. Call this node S. To remove S, simply havethe parent of S change its pointer to point to the right child of S. We know that Shas no left child (because if S did have a left child, S would not be the node with