From 96738dec6cf832f7444197121aaa9972b1a65775 Mon Sep 17 00:00:00 2001 From: e2hang <2099307493@qq.com> Date: Tue, 26 Aug 2025 21:39:35 +0800 Subject: [PATCH] Balance-Tree-Altered --- BinaryTree/BalanceTree/B-Tree/README.MD | 136 ++++++++++++++++++ BinaryTree/BalanceTree/B-Tree/img/erase/1.png | Bin 0 -> 32388 bytes BinaryTree/BalanceTree/B-Tree/img/erase/2.png | Bin 0 -> 28236 bytes BinaryTree/BalanceTree/B-Tree/img/erase/3.png | Bin 0 -> 27032 bytes BinaryTree/BalanceTree/B-Tree/img/erase/4.png | Bin 0 -> 27481 bytes BinaryTree/BalanceTree/B-Tree/img/erase/5.png | Bin 0 -> 25829 bytes BinaryTree/BalanceTree/B-Tree/img/erase/6.png | Bin 0 -> 28899 bytes .../BalanceTree/B-Tree/img/insert/1.png | Bin 0 -> 2599 bytes .../BalanceTree/B-Tree/img/insert/10.png | Bin 0 -> 28461 bytes .../BalanceTree/B-Tree/img/insert/2.png | Bin 0 -> 3114 bytes .../BalanceTree/B-Tree/img/insert/3.png | Bin 0 -> 6633 bytes .../BalanceTree/B-Tree/img/insert/4.png | Bin 0 -> 9038 bytes 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mode 100644 BinaryTree/BalanceTree/B-Tree/img/insert/2.png create mode 100644 BinaryTree/BalanceTree/B-Tree/img/insert/3.png create mode 100644 BinaryTree/BalanceTree/B-Tree/img/insert/4.png create mode 100644 BinaryTree/BalanceTree/B-Tree/img/insert/5.png create mode 100644 BinaryTree/BalanceTree/B-Tree/img/insert/6.png create mode 100644 BinaryTree/BalanceTree/B-Tree/img/insert/7.png create mode 100644 BinaryTree/BalanceTree/B-Tree/img/insert/8.png create mode 100644 BinaryTree/BalanceTree/B-Tree/img/insert/9.png create mode 100644 BinaryTree/BalanceTree/Red-Black Tree/info.jpg diff --git a/BinaryTree/BalanceTree/B-Tree/README.MD b/BinaryTree/BalanceTree/B-Tree/README.MD index 7be146c..d562f4d 100644 --- a/BinaryTree/BalanceTree/B-Tree/README.MD +++ b/BinaryTree/BalanceTree/B-Tree/README.MD @@ -569,6 +569,142 @@ deleteKey(node, k): --- +✅ 对的,你理解完全正确!我帮你梳理一下: + +--- + +### 在 B-树删除时,如果要删除的关键字 `x->keys[i]` 在**内部节点**: + +* **前驱(predecessor)** + + * 就是 `x->children[i]` 这个子树中的最大值。 + * 为什么?因为 `x->children[i]` 里所有关键字都 **小于** `x->keys[i]`,而前驱就是其中最大的。 + * 找法:一路往右孩子走直到叶子,拿到最后一个 key。 + * 伪代码: + + ```cpp + Node* cur = x->children[i]; + while (!cur->isLeaf) cur = cur->children.back(); + T predecessor = cur->keys.back(); + ``` + +* **后继(successor)** + + * 就是 `x->children[i+1]` 这个子树中的最小值。 + * 为什么?因为 `x->children[i+1]` 里所有关键字都 **大于** `x->keys[i]`,而后继就是其中最小的。 + * 找法:一路往左孩子走直到叶子,拿到第一个 key。 + * 伪代码: + + ```cpp + Node* cur = x->children[i+1]; + while (!cur->isLeaf) cur = cur->children.front(); + T successor = cur->keys.front(); + ``` + +--- + +### 删除时的策略 + +* 如果前驱所在子树 `x->children[i]` 有 ≥ t 个 key: + + * 用前驱替代 `x->keys[i]`,再递归删除前驱。 +* 否则如果后继所在子树 `x->children[i+1]` 有 ≥ t 个 key: + + * 用后继替代 `x->keys[i]`,再递归删除后继。 +* 否则(两个子树都只有 t-1 个 key): + + * 把 `x->keys[i]` 下移到左子树,和右子树合并,再递归删除。 + +--- + +📘 **直观理解**: +`x->keys[i]` 在父节点,它左边整个子树都比它小,右边整个子树都比它大。 +所以前驱就是左子树里最大值,后继就是右子树里最小值。 + +--- + +要不要我帮你画一个 **图解例子**(比如 `t=2` 的 B 树,删除一个内部节点的情况),演示「前驱替代」和「后继替代」的区别? + + 如果你需要,我可以帮你 **写一个完整的 C++ B 树删除函数实现**,直接和你已有的 `BTree` 类整合,并附详细注释。 你希望我帮你写吗? + +--- +``` html +总结: +1、删除 +2、查看是否下溢出 +3、下溢出: + a.左右节点可以借(大于t - 1) + 父亲下来,借的成为父亲 + b.不可以借 + 和兄弟合并 + 总结得很到位 👍,我帮你整理一下,更系统一些,方便你以后复习 B-树删除算法。 +``` +--- + +## 🌳 B-树删除算法总结 + +### 1. 删除目标 + +* **在叶子节点找到 key** + 👉 直接删除,不涉及复杂调整。 + +* **在内部节点找到 key** + 👉 不能直接删,而是通过 **前驱 / 后继替代** 或 **合并** 来保持 B-树平衡。 + +--- + +### 2. 删除步骤总览 + +1. **删除目标 key** + + * 在叶子:直接删。 + * 在内部:找前驱或后继替换,再递归删除。 + +2. **检查是否下溢(underflow)** + + * B-树的每个节点至少要有 `t - 1` 个 key(除了根)。 + * 如果某个子节点删除后不足 `t - 1`,则发生下溢。 + +3. **处理下溢** + + * **a. 向兄弟借关键字(兄弟 ≥ t)** + + * 父节点“下放”一个关键字到下溢的节点,兄弟的关键字“上升”到父节点。 + * 保证节点恢复到 ≥ `t - 1`。 + * 📌 这是 **旋转(rotation)** 操作。 + + * **b. 无法借(兄弟也是 t - 1) → 合并** + + * 将下溢节点、父节点中一个关键字、兄弟节点 **合并** 成一个节点。 + * 父节点关键字数减少 1。 + * 如果父节点变空,整棵树高度减 1(根被替换)。 + +--- + +### 3. 前驱 / 后继选择 + +* **前驱**:`x->keys[i]` 的左子树 `x->children[i]` 中的最大值。 +* **后继**:`x->keys[i]` 的右子树 `x->children[i+1]` 中的最小值。 +* 替代规则: + + * 左子树有足够关键字(≥ t),用前驱替代。 + * 否则若右子树有足够关键字(≥ t),用后继替代。 + * 否则左右子树都不够,执行合并。 + +--- + +### 📌 最简总结版(你说的很对) + +1. 删除目标 key。 +2. 如果节点关键字数 ≥ `t - 1`,不用处理。 +3. 如果 < `t - 1`(下溢): + + * **能借** → 借兄弟。 + * **不能借** → 合并兄弟和父节点关键字。 + +--- + +要不要我帮你画一个小例子(`t=2` 的 B-树),演示 **借** 和 **合并** 两种下溢处理,这样你一眼就能看懂? diff --git a/BinaryTree/BalanceTree/B-Tree/img/erase/1.png b/BinaryTree/BalanceTree/B-Tree/img/erase/1.png new file mode 100644 index 0000000000000000000000000000000000000000..4935f4483a10f1a06904eede715662c73e89a6f1 GIT binary patch literal 32388 zcmeFZWmJ@1*ftE~s6&V_v;xv011KO}(w$03w;~`N(nyz-bVvyZ(ls;)(jgLpAR!?Q zA|UdRkI z+*ewZ>N@7!5{arWF)>oZvLNV*h0_%}9e|1qmyUl7Hlx%EK-ynYISW@)l zO+(SzsMlNNblw#~@6%uQ`HV1H>i@pvXziq94~k~Od=Vl4eGx>@72@27ON#xUFX}=# zy&EHOsSEpmKa7^TO=IG}pMnSxQ^%6sQM_%isrP?=5cIkJ`6ZYH`2TtNKPboAo*dpr1?Fr<(K z5A|9U^i1?|t;hT8Gq)~&$<$g7WtdSh&Y!Srm%VN{T27rgJDlAf6j(0~*fta-`g<3$ z?3h>?JxfJR;ow=s3J?^+&qo6WA5~kS-zcR9y?v(RY_UH6&UUmwZG3yX<-)+}6&7?`E7MRUK%2 z+}9uf`{_ymLAs04~kx#67Jt?Ias8hJ~7>5?Ypk? zfusUf_HX~aP%abZJJ^WhwDUN0m(iB6U_45PN{a!9o4mG8qnB+ymt{TuQnMM#$rQ(^ 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