树层高的影响
其中一个是影响对比的次数，每一个节点都存储在磁盘，则每次对比后找下一个节点是一次磁盘寻址，从而影响效率。
多叉树不等于B树
B树是所有叶子结点在同一层的树。
B树性质
- 根结点的儿子树，在[2,M]之间，M指的是树的阶数，就是有几叉。
- 除根结点外，所有非叶子结点的儿子树在M/2(这里要向上取整，2.2取3，但是有些地方好像是不向上取整的)到M之间，例如如果是5阶的B树，那么非叶子结点的儿子树在3~5之间。
- 在叶子结点上，每个叶子结点包含的数据项在L/2(向上取整)到L之间，这里的L一般和M是相等的，但不是必须相等的。
- 当关键字的数量大于L时，就要进行分裂，在父节点上添加关键字，如果父节点的关键字也超过了L，那就继续往上，知道根节点位置，这是唯一增加B树深度的办法。（分裂）
- 当删除关键字的时候，如果数量小于L/2，向父节点借一个节点，并与兄弟节点合并。（借位与合并）
B树与B+树的区别
- B树就是B-树：所有的节点存储数据。（节点占用空间大）
- B+树：叶子结点存储数据，内节点当索引用。（内节点所占空间小）
- B+树更适合做磁盘索引，因为可以把所有的内节点加载到内存中，然后找目标节点快（一次）。而反观B树，只能加载部分节点到内存中，然后进行比较，并在磁盘中寻址，寻址后继续比较，循环往复，在磁盘中查找多次，效率低。
B树相关代码
B树结构
typedef int KEY_VALUE;
typedef struct _btree_node {
KEY_VALUE *keys;
struct _btree_node **childrens;
int num;
int leaf;
} btree_node;
typedef struct _btree {
btree_node *root;
int t;
} btree;
增加B树节点
B树添加都是添加在叶子节点，通过分裂增加层高，中间节点变为父节点。
创建B树节点
btree_node *btree_create_node(int t, int leaf) {
btree_node *node = (btree_node*)calloc(1, sizeof(btree_node));
if (node == NULL) assert(0);
node->leaf = leaf;
node->keys = (KEY_VALUE*)calloc(1, (2*t-1)*sizeof(KEY_VALUE));
node->childrens = (btree_node**)calloc(1, (2*t) * sizeof(btree_node*));
node->num = 0;
return node;
}
创建B树
void btree_create(btree *T, int t) {
T->t = t;
btree_node *x = btree_create_node(t, 1);
T->root = x;
}
分裂
void btree_split_child(btree *T, btree_node *x, int i) {
int t = T->t;
btree_node *y = x->childrens[i];
btree_node *z = btree_create_node(t, y->leaf);
z->num = t - 1;
int j = 0;
for (j = 0;j < t-1;j ++) {
z->keys[j] = y->keys[j+t];
}
if (y->leaf == 0) {
for (j = 0;j < t;j ++) {
z->childrens[j] = y->childrens[j+t];
}
}
y->num = t - 1;
for (j = x->num;j >= i+1;j --) {
x->childrens[j+1] = x->childrens[j];
}
x->childrens[i+1] = z;
for (j = x->num-1;j >= i;j --) {
x->keys[j+1] = x->keys[j];
}
x->keys[i] = y->keys[t-1];
x->num += 1;
}
插入
void btree_insert_nonfull(btree *T, btree_node *x, KEY_VALUE k) {
int i = x->num - 1;
if (x->leaf == 1) {
while (i >= 0 && x->keys[i] > k) {
x->keys[i+1] = x->keys[i];
i --;
}
x->keys[i+1] = k;
x->num += 1;
} else {
while (i >= 0 && x->keys[i] > k) i --;
if (x->childrens[i+1]->num == (2*(T->t))-1) {
btree_split_child(T, x, i+1);
if (k > x->keys[i+1]) i++;
}
btree_insert_nonfull(T, x->childrens[i+1], k);
}
}
void btree_insert(btree *T, KEY_VALUE key) {
//int t = T->t;
btree_node *r = T->root;
if (r->num == 2 * T->t - 1) {
btree_node *node = btree_create_node(T->t, 0);
T->root = node;
node->childrens[0] = r;
btree_split_child(T, node, 0);
int i = 0;
if (node->keys[0] < key) i++;
btree_insert_nonfull(T, node->childrens[i], key);
} else {
btree_insert_nonfull(T, r, key);
}
}
合并
void btree_merge(btree *T, btree_node *node, int idx) {
btree_node *left = node->childrens[idx];
btree_node *right = node->childrens[idx+1];
int i = 0;
/data merge
left->keys[T->t-1] = node->keys[idx];
for (i = 0;i < T->t-1;i ++) {
left->keys[T->t+i] = right->keys[i];
}
if (!left->leaf) {
for (i = 0;i < T->t;i ++) {
left->childrens[T->t+i] = right->childrens[i];
}
}
left->num += T->t;
//destroy right
btree_destroy_node(right);
//node
for (i = idx+1;i < node->num;i ++) {
node->keys[i-1] = node->keys[i];
node->childrens[i] = node->childrens[i+1];
}
node->childrens[i+1] = NULL;
node->num -= 1;
if (node->num == 0) {
T->root = left;
btree_destroy_node(node);
}
}
销毁节点
void btree_delete_key(btree *T, btree_node *node, KEY_VALUE key) {
if (node == NULL) return ;
int idx = 0, i;
while (idx < node->num && key > node->keys[idx]) {
idx ++;
}
if (idx < node->num && key == node->keys[idx]) {
if (node->leaf) {
for (i = idx;i < node->num-1;i ++) {
node->keys[i] = node->keys[i+1];
}
node->keys[node->num - 1] = 0;
node->num--;
if (node->num == 0) { //root
free(node);
T->root = NULL;
}
return ;
} else if (node->childrens[idx]->num >= T->t) {
btree_node *left = node->childrens[idx];
node->keys[idx] = left->keys[left->num - 1];
btree_delete_key(T, left, left->keys[left->num - 1]);
} else if (node->childrens[idx+1]->num >= T->t) {
btree_node *right = node->childrens[idx+1];
node->keys[idx] = right->keys[0];
btree_delete_key(T, right, right->keys[0]);
} else {
btree_merge(T, node, idx);
btree_delete_key(T, node->childrens[idx], key);
}
} else {
btree_node *child = node->childrens[idx];
if (child == NULL) {
printf("Cannot del key = %d\n", key);
return ;
}
if (child->num == T->t - 1) {
btree_node *left = NULL;
btree_node *right = NULL;
if (idx - 1 >= 0)
left = node->childrens[idx-1];
if (idx + 1 <= node->num)
right = node->childrens[idx+1];
if ((left && left->num >= T->t) ||
(right && right->num >= T->t)) {
int richR = 0;
if (right) richR = 1;
if (left && right) richR = (right->num > left->num) ? 1 : 0;
if (right && right->num >= T->t && richR) { //borrow from next
child->keys[child->num] = node->keys[idx];
child->childrens[child->num+1] = right->childrens[0];
child->num ++;
node->keys[idx] = right->keys[0];
for (i = 0;i < right->num - 1;i ++) {
right->keys[i] = right->keys[i+1];
right->childrens[i] = right->childrens[i+1];
}
right->keys[right->num-1] = 0;
right->childrens[right->num-1] = right->childrens[right->num];
right->childrens[right->num] = NULL;
right->num --;
} else { //borrow from prev
for (i = child->num;i > 0;i --) {
child->keys[i] = child->keys[i-1];
child->childrens[i+1] = child->childrens[i];
}
child->childrens[1] = child->childrens[0];
child->childrens[0] = left->childrens[left->num];
child->keys[0] = node->keys[idx-1];
child->num ++;
node->key[idx-1] = left->keys[left->num-1];
left->keys[left->num-1] = 0;
left->childrens[left->num] = NULL;
left->num --;
}
} else if ((!left || (left->num == T->t - 1))
&& (!right || (right->num == T->t - 1))) {
if (left && left->num == T->t - 1) {
btree_merge(T, node, idx-1);
child = left;
} else if (right && right->num == T->t - 1) {
btree_merge(T, node, idx);
}
}
}
btree_delete_key(T, child, key);
}
}
int btree_delete(btree *T, KEY_VALUE key) {
if (!T->root) return -1;
btree_delete_key(T, T->root, key);
return 0;
}
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