[数据结构]B树的基本定义和操作

SilentSlot2026-04-02 9:50

B树

B树的定义

多叉搜索树，每个节点最多有M个子树，M-1个关键字
根节点若非唯一节点，则最少有2个子树，即最少有1个关键字
非根节点也非叶子节点的，最少有ceil(M/2)个子树，ceil(M/2)-1个关键字
叶子节点都在树的同一层
每个节点的关键字从小到大排列，充当划分子树的边界

c 复制代码

// B树的结构体如下：
typedef struct _btree_node {
	int *keys;
	struct _btree_node **childrens;
	int num;  // 有多少个关键字

	int leaf; // 是否是叶子节点
} btree_node;

typedef struct _btree {
	struct _btree_node *root;
	int t;  // ceil(M/2)-1，方便B树的分裂和合并操作
} btree;

B树的节点生成和销毁

c 复制代码

btree_node *btree_node_create(int t, int leaf)
{
	btree_node *node = calloc(1, sizeof(btree_node ));
	node->keys= calloc(2*t-1, sizeof(int));
	node->childrens = calloc(2*t, sizeof(btree_node *));
	node->num = 0;
	node->leaf = leaf;
	return node;
}

void btree_node_destroy(btree_node *node)
{
	free(node->keys);
	free(node->childrens);
	free(node);
}

B树的分裂

c 复制代码

void btree_split_child(btree *T, btree_node *x, int idx) {
	btree_node *y = x->childrens[idx];
	int t = T->t;
	btree_node *z = btree_node_create(t, y->leaf);

	for (int i = 0; i < t - 1; i++) {
		z->keys[i] = y->keys[i+t];
	}
	if (y->leaf == 0) {
		for (int i = 0; i < t; i++) {
			z->childrens[i] = y->chiledrens[i+t];
		}
	}
	z->num = t - 1;
	y->num = t - 1;

	for (int i = x->num; i > idx; i--) {
		x->keys[i] = x->keys[i-1];
		x->childrens[i+1] = x->childrens[i];
	}
	x->keys[idx] = y->keys[t-1];
	x->childrens[idx+1] = z;
	x->num += 1;
}

B树的合并

c 复制代码

void btree_merge(btree *T, btree_node *x, int idx) {
	btree_node *y = x->childrens[idx];
	btree_node *z = x->childrens[idx+1];

	int t = T->t;
	for (int i = 0; i < t - 1; i++) {
		y->keys[i+t] = z->keys[i];
	}
	y->keys[t - 1] = x->keys[idx];

	if (z->leaf == 0) {
		for (int i = 0; i < t; i++) {
			y->childrens[i+t] = z->childrens[i];
		}
	}
	y->num += t;
	btree_node_destroy(z);

	for (int i = idx + 1; i < x->num; i++) {
		x->keys[i-1] = x->keys[i];
		x->childrens[i] = x->childrens[i+1];
	}
	x->num -= 1;
}