二叉搜索树（BST）的插入和删除递归实现

思路

---

二叉搜索树的插入

TreeNode InsertRec(rootNode, key) = 

　　　　if rootNode == NULL, return new Node(key)

　　　　if key >= rootNode.data, rootNode.rightChild = InsertRec(rootNode.rightChild, key)

 　　　　if Key < rootNode.data, rootNode.leftChild = InsertRec(rootNode.leftChild, key)

Context is: currentNode的reference 和要插入的key.

把key插入一个以rootNode为根的子树中去

二叉搜索树的删除

TreeNode DeleteRec(rootNode, key) = 

　　　　if rootNode == NULL, return rootNode

　　　　if key >= rootNode.data, rootNode.rightChild = DeleteRec(rootNode.rightChild, key)

 　　　　if Key < rootNode.data, rootNode.leftChild = DeleteRec(rootNode.leftChild, key)

Context is: currentNode的reference 和要删除的key.

把key从一个以rootNode为根的子树中删去

在base case里，

1. 如果是叶子节点返回空即可，
2. 如果是单子节点，如果左子为空就返回右子。如果右子为空，就返回左子。
3. 如果被删节点有两个孩子，则组要借助util找到中序遍历的后继节点， take 其值，然后再递归删除successor节点。

实现

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        /// <summary>
        /// insert record to BST recursively
        /// </summary>
        /// <param name="root">current root node</param>
        /// <param name="key">the value of the element to be inserted</param>
        public TreeNode<T> InsertRec(TreeNode<T> rootNode, T key)
        {
            if (rootNode == null)
            {
                rootNode = new TreeNode<T>();
                rootNode.data = key;
                return rootNode;
            }

            if (key.CompareTo(rootNode.data) < 0)
            {
                rootNode.leftChild = InsertRec(rootNode.leftChild, key);
            }
            else
            {
                rootNode.rightChild = InsertRec(rootNode.rightChild, key);
            }

            return rootNode;
        }

        public void DeleteKey(T key)
        {
            this.root = DeleteRec(root, key);
        }

        public void InsertKey(T key)
        {
            this.root = InsertRec(root, key);
        }

        /// <summary>
        /// Delete record from BST recursively
        /// </summary>
        /// <param name="rootNode">root node</param>
        /// <param name="Key">value of the element</param>
        /// <returns></returns>
        public TreeNode<T> DeleteRec(TreeNode<T> node, T key)
        {
            if (node == null)
            {
                return null;
            }

            if (key.CompareTo(node.data) < 0)
            {
                node.leftChild = DeleteRec(node.leftChild, key);
            }
            else if (key.CompareTo(node.data) > 0)
            {
                node.rightChild = DeleteRec(node.rightChild, key);
            }
            else // find the node, node is the one to be deleted
            {
                if (node.leftChild == null)
                {
                    return node.rightChild;
                }
                else if (node.rightChild == null)
                {
                    return node.leftChild;
                }
                else
                {
                    // need to handle the root?
                    T value = GetMinValue(node);
                    node.data = value;
                    node.rightChild = DeleteRec(node.rightChild, value);
                }
            }

            return node;
        }

        private T GetMinValue(TreeNode<T> node)
        {
            if (node == null)
            {
                throw new Exception("node is null.");
            }

            if (node.rightChild != null)
            {
                TreeNode<T> temp = node.rightChild;
                while (temp.leftChild != null)
                {
                    temp = temp.leftChild;
                }

                return temp.data;
            }
            else
            {
                throw new Exception("successor node is not found");
            }
        }