b+树详解，代码实例

B+树是一种常用的数据结构，它可以在磁盘上高效地存储和检索数据。B+树是一种平衡树，它的每个非叶子节点可以有多个子节点，而且每个子节点都存储着一定范围内的关键字。

B+树的特点如下：
1. 所有数据都存储在叶子节点中，非叶子节点只存储索引信息。
2. 叶子节点之间是有序的，并且形成了一个链表，可以支持范围查询和顺序访问。
3. 非叶子节点中的索引信息是有序的，并且可以通过索引信息快速定位到对应的叶子节点。
4. B+树的每个节点都至少包含 t 个关键字，其中 t 是一个常数（通常取值为2-4）。

B+树的插入和删除操作比较复杂，需要进行分裂和合并等操作来保证树的平衡性。下面是B+树的插入和删除过程：

插入操作：
1. 如果树为空，则新建一个叶子节点并插入关键字。
2. 如果插入的位置在叶子节点中，则直接插入关键字，并保持叶子节点有序。
3. 如果插入的位置在非叶子节点中，则递归地将关键字插入到对应的子节点中。
4. 如果插入后导致节点关键字数目超过了 t，则进行分裂操作，并将分裂后的两个节点插入到父节点中。

删除操作：
1. 如果要删除的关键字在叶子节点中，则直接删除，并保持叶子节点有序。
2. 如果要删除的关键字在非叶子节点中：
- 如果对应子节点中的关键字数量大于 t，则找到前驱或后继关键字替换，并递归删除替换后的关键字。
- 如果对应子节点中的关键字数量等于 t，则进行合并操作，并递归删除合并后的关键字。
3. 如果删除导致某个节点的关键字数量小于 t/2，则进行合并操作，并将合并后的节点插入到父节点中。

下面是一个简单的B+树实现代码（仅供参考）：

struct Node {
    bool isLeaf;
    int numKeys;
    int keys[MAX_KEYS];
    Node* children[MAX_CHILDREN];
    Node* next;
};

Node* createNode(bool isLeaf) {
    Node* node = new Node;
    node->isLeaf = isLeaf;
    node->numKeys = 0;
    node->next = NULL;
    for (int i = 0; i < MAX_KEYS; i++) {
        node->keys[i] = 0;
    }
    for (int i = 0; i < MAX_CHILDREN; i++) {
        node->children[i] = NULL;
    }
    return node;
}

Node* search(Node* node, int key) {
    if (node == NULL) {
        return NULL;
    }
    int i = 0;
    while (i < node->numKeys && key > node->keys[i]) {
        i++;
    }
    if (node->isLeaf && i < node->numKeys && key == node->keys[i]) {
        return node;
    }
    return search(node->children[i], key);
}

void split(Node* parent, int pos, Node* child) {
    Node* newChild = createNode(child->isLeaf);
    newChild->numKeys = MAX_KEYS / 2;
    for (int i = 0; i < newChild->numKeys; i++) {
        newChild->keys[i] = child->keys[i + MAX_KEYS / 2];
    }
    if (!child->isLeaf) {
        for (int i = 0; i < MAX_CHILDREN / 2; i++) {
            newChild->children[i] = child->children[i + MAX_CHILDREN / 2];
        }
    }
    child->numKeys = MAX_KEYS / 2;
    for (int i = parent->numKeys; i > pos; i--) {
        parent->children[i + 1] = parent->children[i];
        parent->keys[i] = parent->keys[i - 1];
    }
    parent->children[pos + 1] = newChild;
    parent->keys[pos] = newChild->keys;
    parent->numKeys++;
}

void insert(Node*& root, int key) {
    if (root == NULL) {
        root = createNode(true);
        root->keys = key;
        root->numKeys++;
        return;
    }
    Node* node = root;
    Node* parent = NULL;
    int pos = -1;
    while (!node->isLeaf) {
        parent = node;
        pos = -1;
        for (int i = 0; i < node->numKeys; i++) {
            if (key >= node->keys[i]) {
                pos = i;
            } else {
                break;
            }
        }
        if (pos == -1) {
            node = node->children;
        } else {
            node = node->children[pos + 1];
        }
    }
    pos = -1;
    for (int i = 0; i < node->numKeys; i++) {
        if (key == node->keys[i]) {
            return;
        } else if (key > node->keys[i]) {
            pos = i;
        } else {
            break;
        }
    }
    for (int i = node->numKeys; i > pos + 1; i--) {
        node->keys[i] = node->keys[i - 1];
    }
    node->keys[pos + 1] = key;
    node->numKeys++;
    if (node->numKeys == MAX_KEYS) {
        if (parent == NULL) {
            parent = createNode(false);
            root = parent;
            parent->children = node;
        }
        split(parent, pos, node);
    }
}

void merge(Node* parent, int pos, Node* left, Node* right) {
    left->keys[left->numKeys++] = parent->keys[pos];
    for (int i = 0; i < right->numKeys; i++) {
        left->keys[left->numKeys++] = right->keys[i];
    }
    if (!left->isLeaf) {
        for (int i = 0; i < right->numKeys; i++) {
            left->children[left->numKeys + i] = right->children[i];
        }
    }
    left->next = right->next;
    delete right;
    for (int i = pos; i < parent->numKeys - 1; i++) {
        parent->children[i + 1] = parent->children[i + 2];
        parent->keys[i] = parent->keys[i + 1];
    }
    parent->numKeys--;
}

void remove(Node*& root, int key) {
    if (root == NULL) {
        return;
    }
    Node* node = root;
    Node* parent = NULL;
    int pos = -1;
    while (!node->isLeaf) {
        parent = node;
        pos = -1;
        for (int i = 0; i < node->numKeys; i++) {
            if (key >= node->keys[i]) {
                pos = i;
            } else {
                break;
            }
        }
        if (pos == -1) {
            node = node->children;
        } else {
            node = node->children[pos + 1];
        }
    }
    pos = -1;
    for (int i = 0; i < node->numKeys; i++) {
        if (key == node->keys[i]) {
            pos = i;
            break;
        }
    }
    if (pos == -1) {
        return;
    }
    for (int i = pos; i < node->numKeys - 1; i++) {
        node->keys[i] = node->keys[i + 1];
    }
    node->numKeys--;
    if (node == root) { // root is a leaf
        return;
    }
    while (node != root && node != NULL && node->numKeys < MAX_KEYS / 2) {
        int leftPos = -1, rightPos = -1;
        for (int i = 0; i <= parent->numKeys; i++) {
            if (parent->children[i] == node) {
                leftPos = i - 1;
                rightPos = i + 1;
                break;
            }
        }
        Node *leftChild, *rightChild;
        bool flagLeftOverFlow, flagRightOverFlow;
        if (leftPos >= 0) { // has left sibling
            leftChild = parent->children[leftPos];
            flagLeftOverFlow =
                leftChild != NULL && leftChild != NULL &&
                leftChild->numKeys > MAX_KEYS / 2; // whether left sibling can
                                                   // afford one key
            if (!flagLeftOverFlow && leftPos >= 0 && rightPos <= parent->numKeys) { // merge with left sibling
                merge(parent, leftPos, leftChild, node);
                node =
                    leftChild; // after merge, the current position is the new merged child
                continue;
            }
        } else { // no left sibling
            flagLeftOverFlow =
                false; // this flag is set to false to ensure that it will not trigger merge operation with left sibling later
            leftChild =
                NULL; // leftChild is set to NULL to avoid using it later
        }
        if (rightPos <= parent->numKeys) { // has right sibling
            rightChild =
                parent
                    ->children[rightPos]; // after merge operation with left sibling, right sibling could have been moved to the current position
                                           // therefore, rightPos should be calculated again
            flagRightOverFlow =
                rightChild != NULL && rightChild != NULL &&
                rightChild
                    ->numKeys > MAX_KEYS / 2; // whether right sibling can afford one key
            if (!flagRightOverFlow && leftPos >= 0 && rightPos <= parent->numKeys) { // merge with right sibling
                merge(parent, leftPos, node, rightChild);
                node =
                    leftChild; // after merge, the current position is the new merged child
                continue;
            }
        } else { // no right sibling
            flagRightOverFlow =
                false; // this flag is set to false to ensure that it will not trigger merge operation with right sibling later
            rightChild =
                NULL; // rightChild is set to NULL to avoid using it later
        }

        if (flagLeftOverFlow || flagRightOverFlow) { // borrow from sibling and update parent's keys
            if (flagLeftOverFlow &&
                (!flagRightOverFlow ||
                 leftChild
                         ->numKeys >
                     rightChild
                         ->numKeys)) { // prefer borrowing from left sibling
                for (int i =
                         node
                             ->numKeys -
                     1; // move current keys to make room for borrowed key from sibling
                     i >= 0; --i) {
                    node
                        ->keys[i +
                              1] =
                        node
                            ->keys[i];
                }
                if (!node
                         ->isLeaf) { // move children accordingly for non-leaf nodes
                    for (int i =
                             node->
                             numKeys -
                         1; i >=
                             0; --i) {
                        node
                            ->children[i +
                                       2] =
                            node->
                            children[i +
                                     1];
                    }
                    node
                        ->children =
                        leftChild->
                        children[leftChild->
                                  numKeys +
                                  1];
                    ++node->
                        numKeys;

                    leftChild->
                        children[leftChild->
                                  numKeys +
                                  1] =
                        NULL;

                    --leftChild->
                        numKeys;

                    parent->
                        keys[leftPos] =
                        leftChild->
                        keys[leftChild->
                                 numKeys -
                             1];

                    leftChild->
                        keys[leftChild->
                                 numKeys -
                             1] =
                        0;

                    continue;

                } else { // leaf nodes need to update next pointer as well
                    ++node->
                        numKeys;

                    int newKeyIndex =
                        --leftChild->
                        numKeys;

                    parent->
                        keys[leftPos] =
                        leftChild->
                        keys[newKeyIndex];

                    ++node->
                        numKeys;

                    node
                        ->next =
                        leftChild->
                        next;

                    leftChild->
                        next =
                        node;

                    continue;

                }

            } else { // borrow from right sibling or its children
                if (!node->
                         isLeaf) { // move children accordingly for non-leaf nodes

                    ++node->
                        numKeys;

                    int newKeyIndex =
                        ++node->
                        numKeys;

                    if (newKeyIndex >= MAX_KEYS)
                        newKeyIndex--;

                    node
                        ->children[newKeyIndex +
                                    1] =
                        rightChild->
                        children;

                    parent->
                        keys[rightPos -
                             1] =
                        rightChild->
                        keys;

                    ++rightChild->
                        numKeys;

                    for (int k =
                             0; k <
                         rightChild->
                         numKeys -
                         1; ++k)
                        rightChild->
                            keys[k] =
                            rightChild->
                            keys[k +
                                 1];

                    for (int k =
                             0; k <
                         rightChild->
                         numKeys +
                         1; ++k)
                        rightChild->
                            children[k] =
                            rightChild->
                            children[k +
                                     1];

                    --rightChild->
                        numKeys;

                    continue;

                } else { // leaf nodes need to update next pointer as well

                    ++node->
                        numKeys;

                    int newKeyIndex =
                        ++node->
                        numKeys;

                    if (newKeyIndex >= MAX_KEYS)
                        newKeyIndex--;

                    node
                        ->keys[newKeyIndex -
                                1] =
                        rightChild->
                        keys;

                    ++rightChild->
                        numKeys;

                    for (int k =
                             0; k <
                         rightChild->
                         numKeys -
                         1; ++k)
                        rightChild->
                            keys[k] =
                            rightChild->
                            keys[k +
                                 1];

                    --rightChild->
                        numKeys;

                    parent->
                        keys[rightPos -
                             1] =
                        rightChild->
                        keys;

                    continue;

                }

            }

        }

        break;

    }

}