b树删除操作需要考虑节点所在位置和平衡，并且很有可能会发生下溢的情况。当一个节点包含的子节点数量少于它应该持有的最小数量时，就会发生下溢。
图文展示B树删除操作原理
在不影响平衡情况下。
下溢情况。
删除内部节点。
Python实现B树删除操作
# B树节点class BTreeNode: def __init__(self, leaf=False): self.leaf = leaf self.keys = [] self.child = []class BTree: def __init__(self, t): self.root = BTreeNode(True) self.t = t # 插入元素 def insert(self, k): root = self.root if len(root.keys) == (2 * self.t) - 1: temp = BTreeNode() self.root = temp temp.child.insert(0, root) self.split_child(temp, 0) self.insert_non_full(temp, k) else: self.insert_non_full(root, k) def insert_non_full(self, x, k): i = len(x.keys) - 1 if x.leaf: x.keys.append((None, None)) while i >= 0 and k[0] = 0 and k[0] x.keys[i][0]: i += 1 self.insert_non_full(x.child[i], k) # 分开子节点 def split_child(self, x, i): t = self.t y = x.child[i] z = BTreeNode(y.leaf) x.child.insert(i + 1, z) x.keys.insert(i, y.keys[t - 1]) z.keys = y.keys[t: (2 * t) - 1] y.keys = y.keys[0: t - 1] if not y.leaf: z.child = y.child[t: 2 * t] y.child = y.child[0: t - 1] # 删除节点 def delete(self, x, k): t = self.t i = 0 while i x.keys[i][0]: i += 1 if x.leaf: if i < len(x.keys) and x.keys[i][0] == k[0]: x.keys.pop(i) return return if i = t: self.delete(x.child[i], k) else: if i != 0 and i + 2 = t: self.delete_sibling(x, i, i - 1) elif len(x.child[i + 1].keys) >= t: self.delete_sibling(x, i, i + 1) else: self.delete_merge(x, i, i + 1) elif i == 0: if len(x.child[i + 1].keys) >= t: self.delete_sibling(x, i, i + 1) else: self.delete_merge(x, i, i + 1) elif i + 1 == len(x.child): if len(x.child[i - 1].keys) >= t: self.delete_sibling(x, i, i - 1) else: self.delete_merge(x, i, i - 1) self.delete(x.child[i], k) # 删除节点 def delete_internal_node(self, x, k, i): t = self.t if x.leaf: if x.keys[i][0] == k[0]: x.keys.pop(i) return return if len(x.child[i].keys) >= t: x.keys[i] = self.delete_predecessor(x.child[i]) return elif len(x.child[i + 1].keys) >= t: x.keys[i] = self.delete_successor(x.child[i + 1]) return else: self.delete_merge(x, i, i + 1) self.delete_internal_node(x.child[i], k, self.t - 1) # 删除前节点 def delete_predecessor(self, x): if x.leaf: return x.pop() n = len(x.keys) - 1 if len(x.child[n].keys) >= self.t: self.delete_sibling(x, n + 1, n) else: self.delete_merge(x, n, n + 1) self.delete_predecessor(x.child[n]) # 删除继任节点 def delete_successor(self, x): if x.leaf: return x.keys.pop(0) if len(x.child[1].keys) >= self.t: self.delete_sibling(x, 0, 1) else: self.delete_merge(x, 0, 1) self.delete_successor(x.child[0]) def delete_merge(self, x, i, j): cnode = x.child[i] if j > i: rsnode = x.child[j] cnode.keys.append(x.keys[i]) for k in range(len(rsnode.keys)): cnode.keys.append(rsnode.keys[k]) if len(rsnode.child) > 0: cnode.child.append(rsnode.child[k]) if len(rsnode.child) > 0: cnode.child.append(rsnode.child.pop()) new = cnode x.keys.pop(i) x.child.pop(j) else: lsnode = x.child[j] lsnode.keys.append(x.keys[j]) for i in range(len(cnode.keys)): lsnode.keys.append(cnode.keys[i]) if len(lsnode.child) > 0: lsnode.child.append(cnode.child[i]) if len(lsnode.child) > 0: lsnode.child.append(cnode.child.pop()) new = lsnode x.keys.pop(j) x.child.pop(i) if x == self.root and len(x.keys) == 0: self.root = new # 删除同一级的其他子节点 def delete_sibling(self, x, i, j): cnode = x.child[i] if i 0: cnode.child.append(rsnode.child[0]) rsnode.child.pop(0) rsnode.keys.pop(0) else: lsnode = x.child[j] cnode.keys.insert(0, x.keys[i - 1]) x.keys[i - 1] = lsnode.keys.pop() if len(lsnode.child) > 0: cnode.child.insert(0, lsnode.child.pop()) # 输出B树 def print_tree(self, x, l=0): print("Level ", l, " ", len(x.keys), end=":") for i in x.keys: print(i, end=" ") print() l += 1 if len(x.child) > 0: for i in x.child: self.print_tree(i, l)B = BTree(3)for i in range(10): B.insert((i, 2 * i))B.print_tree(B.root)B.delete(B.root, (8,))print("n")B.print_tree(B.root)
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