B-Tree

2025-08-26 19:21:23 +08:00
parent fe9d51456b
commit 3eb93c256c
3 changed files with 434 additions and 1 deletions
--- a/BinaryTree/BalanceTree/B-Tree/B-Tree-By-AI.h
+++ b/BinaryTree/BalanceTree/B-Tree/B-Tree-By-AI.h
@@ -0,0 +1,241 @@
+#include <iostream>
+#include <vector>
+using namespace std;
+
+template <class T>
+struct Node {
+    bool leaf;                // 是否为叶子节点
+    vector<T> keys;           // 存储关键字
+    vector<Node*> children;   // 孩子节点指针
+
+    Node(bool _leaf) : leaf(_leaf) {}
+};
+
+template <class T>
+class BTree {
+public:
+    BTree(int _t) : t(_t), root(nullptr) {}
+
+    void insert(const T& key);   // 插入关键字
+    void remove(const T& key);   // 删除关键字
+    void traverse() { traverse(root); cout << endl; } // 中序遍历
+
+private:
+    Node<T>* root;
+    int t; // 最小度数
+
+    // ----------- 插入相关辅助函数 -----------
+    void insertNonFull(Node<T>* node, const T& key); // 在非满节点插入
+    void splitChild(Node<T>* parent, int idx);      // 分裂子节点
+
+    // ----------- 删除相关辅助函数 -----------
+    void remove(Node<T>* node, const T& key);        // 递归删除
+    void removeFromLeaf(Node<T>* node, int idx);    // 从叶子节点删除
+    void removeFromNonLeaf(Node<T>* node, int idx); // 从非叶子节点删除
+    T getPredecessor(Node<T>* node, int idx);       // 找前驱
+    T getSuccessor(Node<T>* node, int idx);         // 找后继
+    void fill(Node<T>* node, int idx);             // 保证孩子节点至少 t-1 个关键字
+    void borrowFromPrev(Node<T>* node, int idx);   // 从左兄弟借一个关键字
+    void borrowFromNext(Node<T>* node, int idx);   // 从右兄弟借一个关键字
+    void merge(Node<T>* node, int idx);            // 与右兄弟合并
+
+    // ----------- 遍历辅助函数 -----------
+    void traverse(Node<T>* node);
+};
+
+// ---------------------- 插入相关 ----------------------
+template<class T>
+void BTree<T>::insert(const T& key) {
+    if (!root) { // 树空
+        root = new Node<T>(true);
+        root->keys.push_back(key);
+    } else {
+        // 根节点满，先分裂根节点
+        if (root->keys.size() == 2*t - 1) {
+            Node<T>* s = new Node<T>(false);
+            s->children.push_back(root);
+            splitChild(s, 0);
+            root = s;
+        }
+        insertNonFull(root, key); // 插入非满节点
+    }
+}
+
+template<class T>
+void BTree<T>::insertNonFull(Node<T>* node, const T& key) {
+    int i = node->keys.size() - 1;
+    if (node->leaf) { // 叶子节点直接插入
+        node->keys.push_back(key);
+        while (i >= 0 && node->keys[i] > key) {
+            node->keys[i+1] = node->keys[i];
+            i--;
+        }
+        node->keys[i+1] = key;
+    } else { // 内部节点
+        while (i >= 0 && node->keys[i] > key) i--;
+        i++;
+        if (node->children[i]->keys.size() == 2*t-1) {
+            splitChild(node, i); // 分裂满子节点
+            if (node->keys[i] < key) i++;
+        }
+        insertNonFull(node->children[i], key);
+    }
+}
+
+template<class T>
+void BTree<T>::splitChild(Node<T>* parent, int idx) {
+    Node<T>* y = parent->children[idx];
+    Node<T>* z = new Node<T>(y->leaf);
+
+    // 把 y 的后 t-1 个关键字移动到 z
+    for (int j = 0; j < t-1; j++) z->keys.push_back(y->keys[j+t]);
+    if (!y->leaf) {
+        for (int j = 0; j < t; j++) z->children.push_back(y->children[j+t]);
+        y->children.resize(t);
+    }
+
+    y->keys.resize(t-1); // 保留前 t-1 个关键字
+    parent->children.insert(parent->children.begin()+idx+1, z);
+    parent->keys.insert(parent->keys.begin()+idx, y->keys[t-1]); // 中间关键字上移
+}
+
+// ---------------------- 删除相关 ----------------------
+template<class T>
+void BTree<T>::remove(const T& key) {
+    if (!root) return;
+    remove(root, key);
+    // 如果根节点被删除后没有关键字，调整 root
+    if (root->keys.empty()) {
+        Node<T>* oldRoot = root;
+        if (root->leaf) root = nullptr;
+        else root = root->children[0];
+        delete oldRoot;
+    }
+}
+
+template<class T>
+void BTree<T>::remove(Node<T>* node, const T& key) {
+    int idx = 0;
+    while (idx < node->keys.size() && node->keys[idx] < key) idx++;
+
+    if (idx < node->keys.size() && node->keys[idx] == key) {
+        if (node->leaf) removeFromLeaf(node, idx);         // 情况1: 叶子节点直接删除
+        else removeFromNonLeaf(node, idx);                // 情况2: 内部节点
+    } else {
+        if (node->leaf) return; // key不存在
+        bool flag = (idx == node->keys.size());
+        // 如果孩子节点关键字不足 t 个，需要调整
+        if (node->children[idx]->keys.size() < t) fill(node, idx);
+        if (flag && idx > node->keys.size()) remove(node->children[idx-1], key);
+        else remove(node->children[idx], key);
+    }
+}
+
+template<class T>
+void BTree<T>::removeFromLeaf(Node<T>* node, int idx) {
+    // 叶子节点直接删除关键字
+    node->keys.erase(node->keys.begin() + idx);
+}
+
+template<class T>
+void BTree<T>::removeFromNonLeaf(Node<T>* node, int idx) {
+    T k = node->keys[idx];
+    // 如果左孩子关键字 >= t，用前驱替换
+    if (node->children[idx]->keys.size() >= t) {
+        T pred = getPredecessor(node, idx);
+        node->keys[idx] = pred;
+        remove(node->children[idx], pred);
+    }
+    // 如果右孩子关键字 >= t，用后继替换
+    else if (node->children[idx+1]->keys.size() >= t) {
+        T succ = getSuccessor(node, idx);
+        node->keys[idx] = succ;
+        remove(node->children[idx+1], succ);
+    }
+    // 左右孩子都不足 t，则合并孩子
+    else {
+        merge(node, idx);
+        remove(node->children[idx], k);
+    }
+}
+
+template<class T>
+T BTree<T>::getPredecessor(Node<T>* node, int idx) {
+    Node<T>* cur = node->children[idx];
+    while (!cur->leaf) cur = cur->children.back();
+    return cur->keys.back();
+}
+
+template<class T>
+T BTree<T>::getSuccessor(Node<T>* node, int idx) {
+    Node<T>* cur = node->children[idx+1];
+    while (!cur->leaf) cur = cur->children[0];
+    return cur->keys[0];
+}
+
+template<class T>
+void BTree<T>::fill(Node<T>* node, int idx) {
+    // 尝试从兄弟借关键字，如果不够则合并
+    if (idx != 0 && node->children[idx-1]->keys.size() >= t)
+        borrowFromPrev(node, idx);
+    else if (idx != node->keys.size() && node->children[idx+1]->keys.size() >= t)
+        borrowFromNext(node, idx);
+    else {
+        if (idx != node->keys.size()) merge(node, idx);
+        else merge(node, idx-1);
+    }
+}
+
+template<class T>
+void BTree<T>::borrowFromPrev(Node<T>* node, int idx) {
+    Node<T>* child = node->children[idx];
+    Node<T>* sibling = node->children[idx-1];
+
+    // 将父节点关键字下移到 child 前面
+    child->keys.insert(child->keys.begin(), node->keys[idx-1]);
+    if (!child->leaf) child->children.insert(child->children.begin(), sibling->children.back());
+    node->keys[idx-1] = sibling->keys.back();
+    sibling->keys.pop_back();
+    if (!sibling->leaf) sibling->children.pop_back();
+}
+
+template<class T>
+void BTree<T>::borrowFromNext(Node<T>* node, int idx) {
+    Node<T>* child = node->children[idx];
+    Node<T>* sibling = node->children[idx+1];
+
+    // 将父节点关键字下移到 child 后面
+    child->keys.push_back(node->keys[idx]);
+    if (!child->leaf) child->children.push_back(sibling->children[0]);
+    node->keys[idx] = sibling->keys[0];
+    sibling->keys.erase(sibling->keys.begin());
+    if (!sibling->leaf) sibling->children.erase(sibling->children.begin());
+}
+
+template<class T>
+void BTree<T>::merge(Node<T>* node, int idx) {
+    Node<T>* child = node->children[idx];
+    Node<T>* sibling = node->children[idx+1];
+
+    // 将父节点关键字和 sibling 的关键字都移入 child
+    child->keys.push_back(node->keys[idx]);
+    child->keys.insert(child->keys.end(), sibling->keys.begin(), sibling->keys.end());
+    if (!child->leaf) child->children.insert(child->children.end(), sibling->children.begin(), sibling->children.end());
+
+    // 删除父节点中的关键字和 sibling 指针
+    node->keys.erase(node->keys.begin() + idx);
+    node->children.erase(node->children.begin() + idx + 1);
+    delete sibling;
+}
+
+// ---------------------- 遍历 ----------------------
+template<class T>
+void BTree<T>::traverse(Node<T>* node) {
+    if (!node) return;
+    int i;
+    for (i = 0; i < node->keys.size(); i++) {
+        if (!node->leaf) traverse(node->children[i]);
+        cout << node->keys[i] << " ";
+    }
+    if (!node->leaf) traverse(node->children[i]);
+}
--- a/BinaryTree/BalanceTree/B-Tree/BTree.h
+++ b/BinaryTree/BalanceTree/B-Tree/BTree.h
@@ -131,5 +131,169 @@ inline bool BTree<T>::search(Node<T>* _node, const T& _key)
 template<class T>
 inline void BTree<T>::erase(const T& _key)
 {
-	//1-ɾ<><C9BE>Ҷ<EFBFBD><EFBFBD>
+    if (!root) return; // <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֱ<EFBFBD>ӷ<EFBFBD><EFBFBD><EFBFBD>
+
+    // <20><>һ<EFBFBD><D2BB><EFBFBD>ݹ<EFBFBD> lambda <20><>ʵ<EFBFBD><CAB5>ɾ<EFBFBD><C9BE><EFBFBD>߼<EFBFBD>
+    auto remove = [&](auto&& self, Node<T>* node, const T& key) -> void {
+        int idx = 0;
+        // <20>ҵ<EFBFBD><D2B5><EFBFBD>һ<EFBFBD><D2BB> >= key <20><>λ<EFBFBD><CEBB>
+        while (idx < node->keys.size() && node->keys[idx] < key) idx++;
+
+        // ==============================================================
+        // <20><><EFBFBD><EFBFBD> 1<><31>key <20><><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD>ǰ<EFBFBD>ڵ<EFBFBD> node->keys[idx]
+        // ==============================================================
+        if (idx < node->keys.size() && node->keys[idx] == key) {
+            // ---- 1.1 <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ǰ<EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD>Ҷ<EFBFBD>ӣ<EFBFBD>ֱ<EFBFBD><D6B1>ɾ<EFBFBD><C9BE><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
+            if (node->isLeaf) {
+                node->keys.erase(node->keys.begin() + idx);
+            }
+            // ---- 1.2 <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ڲ<EFBFBD><DAB2>ڵ㣨<DAB5><E3A3A8>Ҷ<EFBFBD>ӣ<EFBFBD>
+            else {
+                Node<T>* leftChild = node->children[idx];
+                Node<T>* rightChild = node->children[idx + 1];
+
+                // <20><> <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> t <20><><EFBFBD>ؼ<EFBFBD><D8BC>֣<EFBFBD><D6A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ã<EFBFBD>
+                if (leftChild->keys.size() >= t) {
+                    // <20>ҵ<EFBFBD>ǰ<EFBFBD><C7B0><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ұ<EFBFBD>Ԫ<EFBFBD>أ<EFBFBD>
+                    Node<T>* cur = leftChild;
+                    while (!cur->isLeaf) cur = cur->children.back();
+                    T pred = cur->keys.back();
+                    // <20><>ǰ<EFBFBD><C7B0><EFBFBD>滻<EFBFBD><E6BBBB>ǰ key
+                    node->keys[idx] = pred;
+                    // <20>ݹ<EFBFBD>ȥ<EFBFBD><C8A5><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɾ<EFBFBD><C9BE>ǰ<EFBFBD><C7B0>
+                    self(self, leftChild, pred);
+                }
+                // <20><> <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> t <20><><EFBFBD>ؼ<EFBFBD><D8BC>֣<EFBFBD><D6A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ã<EFBFBD>
+                else if (rightChild->keys.size() >= t) {
+                    // <20>ҵ<EFBFBD><D2B5><EFBFBD><EFBFBD>̣<EFBFBD><CCA3><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ԫ<EFBFBD>أ<EFBFBD>
+                    Node<T>* cur = rightChild;
+                    while (!cur->isLeaf) cur = cur->children.front();
+                    T succ = cur->keys.front();
+                    // <20>ú<EFBFBD><C3BA><EFBFBD><EFBFBD>滻<EFBFBD><E6BBBB>ǰ key
+                    node->keys[idx] = succ;
+                    // <20>ݹ<EFBFBD>ȥ<EFBFBD><C8A5><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɾ<EFBFBD><C9BE><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
+                    self(self, rightChild, succ);
+                }
+                // <20><> <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֻ<EFBFBD><D6BB> t-1 <20><><EFBFBD>ؼ<EFBFBD><D8BC>֣<EFBFBD><D6A3>޷<EFBFBD><DEB7><EFBFBD> <20><> <20>ϲ<EFBFBD>
+                else {
+                    // <20><> key <20>³<EFBFBD><C2B3><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
+                    leftChild->keys.push_back(node->keys[idx]);
+                    // <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> keys ȫ<><C8AB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
+                    leftChild->keys.insert(leftChild->keys.end(),
+                        rightChild->keys.begin(), rightChild->keys.end());
+                    // <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ҷ<EFBFBD>ӣ<EFBFBD><D3A3><EFBFBD>Ҫ<EFBFBD><D2AA><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> children <20><><EFBFBD><EFBFBD>ȥ
+                    if (!leftChild->isLeaf) {
+                        leftChild->children.insert(leftChild->children.end(),
+                            rightChild->children.begin(), rightChild->children.end());
+                    }
+                    // ɾ<><C9BE><EFBFBD><EFBFBD><EFBFBD>ڵ<EFBFBD><DAB5>е<EFBFBD> key <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ָ<EFBFBD><D6B8>
+                    node->keys.erase(node->keys.begin() + idx);
+                    node->children.erase(node->children.begin() + idx + 1);
+                    delete rightChild;
+                    // <20><><EFBFBD><EFBFBD><EFBFBD>ںϲ<DABA><CFB2><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɾ<EFBFBD><C9BE> key
+                    self(self, leftChild, key);
+                }
+            }
+        }
+        // ==============================================================
+        // <20><><EFBFBD><EFBFBD> 2<><32>key <20><><EFBFBD>ڵ<EFBFBD>ǰ<EFBFBD>ڵ<EFBFBD> <20><> ȥ<><C8A5><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
+        // ==============================================================
+        else {
+            if (node->isLeaf) return; // <20><>Ҷ<EFBFBD>ӻ<EFBFBD>û<EFBFBD>ҵ<EFBFBD><D2B5><EFBFBD>˵<EFBFBD><CBB5><EFBFBD><EFBFBD><EFBFBD><EFBFBD>û<EFBFBD>У<EFBFBD>ֱ<EFBFBD>ӷ<EFBFBD><D3B7><EFBFBD>
+
+            bool flag = (idx == node->keys.size()); // key <20><><EFBFBD><EFBFBD><EFBFBD>нڵ㶼<DAB5><E3B6BC> <20><> <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><D2BB><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
+            Node<T>* child = node->children[idx];
+
+            // ----------------------------------------------------------
+            // ɾ<><C9BE>֮ǰҪ<C7B0><D2AA>֤ child <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD> t <20><><EFBFBD>ؼ<EFBFBD><D8BC>֣<EFBFBD><D6A3><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ɾ<EFBFBD>겻<EFBFBD>Ϸ<EFBFBD><CFB7><EFBFBD>
+            // ----------------------------------------------------------
+            if (child->keys.size() < t) {
+                Node<T>* leftSibling = (idx > 0 ? node->children[idx - 1] : nullptr);
+                Node<T>* rightSibling = (idx < node->keys.size() ? node->children[idx + 1] : nullptr);
+
+                // <20><> <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֵܴ<D6B5><DCB4><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> >= t <20><><EFBFBD>ؼ<EFBFBD><D8BC><EFBFBD> <20><> <20><><EFBFBD><EFBFBD><EFBFBD>ֵܽ<D6B5>
+                if (leftSibling && leftSibling->keys.size() >= t) {
+                    // <20><><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><D2BB> key <20><> child
+                    child->keys.insert(child->keys.begin(), node->keys[idx - 1]);
+                    // <20><><EFBFBD>ֵ<EFBFBD><D6B5><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><D2BB> key <20><><EFBFBD>Ƶ<EFBFBD><C6B5><EFBFBD><EFBFBD>ڵ<EFBFBD>
+                    node->keys[idx - 1] = leftSibling->keys.back();
+                    leftSibling->keys.pop_back();
+                    // <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӽڵ㣬<DAB5><E3A3AC><EFBFBD><EFBFBD><EFBFBD>˵<EFBFBD><CBB5><EFBFBD> children
+                    if (!leftSibling->isLeaf) {
+                        child->children.insert(child->children.begin(), leftSibling->children.back());
+                        leftSibling->children.pop_back();
+                    }
+                }
+                // <20><> <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֵܴ<D6B5><DCB4><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> >= t <20><><EFBFBD>ؼ<EFBFBD><D8BC><EFBFBD> <20><> <20><><EFBFBD><EFBFBD><EFBFBD>ֵܽ<D6B5>
+                else if (rightSibling && rightSibling->keys.size() >= t) {
+                    // <20><><EFBFBD>ڵ<EFBFBD><DAB5><EFBFBD><EFBFBD><EFBFBD>һ<EFBFBD><D2BB> key <20><> child
+                    child->keys.push_back(node->keys[idx]);
+                    // <20><><EFBFBD>ֵܵ<D6B5>һ<EFBFBD><D2BB> key <20><><EFBFBD>Ƶ<EFBFBD><C6B5><EFBFBD><EFBFBD>ڵ<EFBFBD>
+                    node->keys[idx] = rightSibling->keys.front();
+                    rightSibling->keys.erase(rightSibling->keys.begin());
+                    // <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ӽڵ㣬<DAB5><E3A3AC><EFBFBD><EFBFBD><EFBFBD>˵<EFBFBD><CBB5><EFBFBD> children
+                    if (!rightSibling->isLeaf) {
+                        child->children.push_back(rightSibling->children.front());
+                        rightSibling->children.erase(rightSibling->children.begin());
+                    }
+                }
+                // <20><> <20><><EFBFBD><EFBFBD><EFBFBD>ֵܶ<D6B5><DCB6><EFBFBD><EFBFBD><EFBFBD> t <20><> <20>ϲ<EFBFBD>
+                else {
+                    if (leftSibling) {
+                        // <20>Ѹ<EFBFBD><D1B8>ڵ<EFBFBD>һ<EFBFBD><D2BB> key <20><><EFBFBD>Ƶ<EFBFBD><C6B5><EFBFBD><EFBFBD>ֵ<EFBFBD>
+                        leftSibling->keys.push_back(node->keys[idx - 1]);
+                        // <20><> child <20><> keys ȫ<><C8AB><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ֵ<EFBFBD>
+                        leftSibling->keys.insert(leftSibling->keys.end(),
+                            child->keys.begin(), child->keys.end());
+                        if (!child->isLeaf) {
+                            leftSibling->children.insert(leftSibling->children.end(),
+                                child->children.begin(), child->children.end());
+                        }
+                        // <20><><EFBFBD>ڵ<EFBFBD>ɾ<EFBFBD><C9BE> key <20><> child ָ<><D6B8>
+                        node->keys.erase(node->keys.begin() + idx - 1);
+                        node->children.erase(node->children.begin() + idx);
+                        delete child;
+                        child = leftSibling;
+                    }
+                    else if (rightSibling) {
+                        // <20>Ѹ<EFBFBD><D1B8>ڵ<EFBFBD>һ<EFBFBD><D2BB> key <20><><EFBFBD>Ƶ<EFBFBD> child
+                        child->keys.push_back(node->keys[idx]);
+                        // <20><><EFBFBD><EFBFBD><EFBFBD>ֵܵ<D6B5> keys ȫ<><C8AB><EFBFBD><EFBFBD><EFBFBD><EFBFBD> child
+                        child->keys.insert(child->keys.end(),
+                            rightSibling->keys.begin(), rightSibling->keys.end());
+                        if (!rightSibling->isLeaf) {
+                            child->children.insert(child->children.end(),
+                                rightSibling->children.begin(), rightSibling->children.end());
+                        }
+                        node->keys.erase(node->keys.begin() + idx);
+                        node->children.erase(node->children.begin() + idx + 1);
+                        delete rightSibling;
+                    }
+                }
+            }
+            // ----------------------------------------------------------
+            // <20><>֤ child <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD> t <20><> key <20>󣬵ݹ<F3A3ACB5><DDB9><EFBFBD>ȥ
+            // ----------------------------------------------------------
+            if (flag && idx > node->keys.size())
+                self(self, node->children[idx - 1], key);
+            else
+                self(self, node->children[idx], key);
+        }
+        };
+
+    // <20><><EFBFBD><EFBFBD>ɾ<EFBFBD><C9BE><EFBFBD>߼<EFBFBD>
+    remove(remove, root, _key);
+
+    // ==============================================================
+    // ɾ<><C9BE><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>ܵ<EFBFBD><DCB5>¸<EFBFBD><C2B8>ڵ<EFBFBD>Ϊ<EFBFBD><CEAA>
+    // <20><>Ҫ<EFBFBD><D2AA><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> root û<><C3BB> key <20><>
+    // ==============================================================
+    if (root->keys.empty()) {
+        Node<T>* oldRoot = root;
+        // <20><><EFBFBD><EFBFBD> root <20><>Ҷ<EFBFBD><D2B6> <20><> <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ϊ<EFBFBD><CEAA>
+        if (root->isLeaf) root = nullptr;
+        else root = root->children[0]; // root <20><>Ψһ<CEA8><D2BB><EFBFBD>ӳ<EFBFBD>Ϊ<EFBFBD>µĸ<C2B5>
+        delete oldRoot;
+    }
 }
+
--- a/BinaryTree/BalanceTree/B-Tree/main.cpp
+++ b/BinaryTree/BalanceTree/B-Tree/main.cpp
@@ -16,6 +16,10 @@ int main() {
 	a.insert(11);a.display();
 	a.insert(15);a.display();
 	a.insert(17);a.display();
+
+	a.erase(7);a.display();
+	a.erase(10);a.display();
+	a.erase(1);a.display();
 	return 0;
 }
 /*
@@ -70,4 +74,28 @@ int main() {
 6 7 8
 9 10
 11 15 17
+
+
+1 2
+3 4 5
+
+6 8
+9 10
+11 15 17
+
+
+1 2
+3 4 5
+
+6 8
+9 11
+15 17
+
+
+2
+3 4 5
+6 8
+
+9 11
+15 17
 */