B树一个Node可以有N个key, N+1个下级Node， 二叉树就是简化版，一个key两个下级node

2-3树和2-3-4树的区不大，2-3树在插入时先找到叶子节点（没有子节点），然后插入，过程中如果已经是3Node(2 key）就分裂，向上冒泡，一直可能冒泡到顶上。

2-3-4树则在向下找叶子节点时就做调整，把4Node(3 key)提前分裂掉，为下级节点腾出空间，所以叶子节点插入后不会不停向上冒泡。

2-3-4树冗余更大，如果不提前分裂就是2-3树

红黑树是2-3-4树的2节点表示，采用左倾和旋转来简化和冒泡，Rober Segwick的ppt很经典

B+树感觉都是数据库中数据和索引的关系。索引可以没有，有了是用来加速某种查找。

在数据增删时索引维护时个问题。

顺便提下skip list, 对排序好的数据提取N个做索引，再对N个做同样采样索引，重复向上。对于大量数据很好理解，也很容易实现。索引和数据分开。

2-3树删除算法比较复杂，这里没实现。仅仅实现了插入和搜索，测试结果：

 

E A R C H X M P L 

A C E H L M P R X 

 (A 

 C) 

 (E) 

 (H 

 L) 

(M) 

 (P) 

 (R) 

 (X) 

16 7 15 15 18 3 6 15 5 11 9 12 7 10 19 18 12 14 2 12 

2 3 5 6 7 9 10 11 12 14 15 16 18 19 

 (2) 

 (3 

 (5) 

 6) 

 (7) 

(9 

 (10) 

 (11) 

 (12 

 14) 

15) 

 (16) 

 (18) 

 (19) 

#include <stdio.h>
#include <stdlib.h>
#include <errno.h>
#include <assert.h>
#include <string.h>

typedef int T;

typedef struct TreeNodeT {
	T v; //value left
	T v2; //value right
	int twins;
	struct TreeNodeT *up;
	struct TreeNodeT *left;
	struct TreeNodeT *center; //center if twins node
	struct TreeNodeT *right;
} TreeNode;

typedef int Callback(TreeNode *node, int n, int level, void *ctx); //return stop or not

TreeNode *tree_del(TreeNode *node, T v); //TODO

TreeNode *tree_top(TreeNode *e) {
	TreeNode *top = NULL;
	while (e && e->up)
		top = e->up;
	return top;
}

int tree_callback_locate(TreeNode *node, int n, int level, void *ctx) {
	T v = n == 0 ? node->v : node->v2;
	T t = (T) ctx;
	return v == t;
}

int tree_callback_verify(TreeNode *node, int n, int level, void *ctx) {
	assert(ctx);
	T v = n == 0 ? node->v : node->v2;
	T *last = (T *) ctx;
	assert(v > *last);
	*last = v;
	return 0;
}

int tree_callback_print_h(TreeNode *node, int n, int level, void *ctx) {
	printf("%d ", n == 0 ? node->v : node->v2);
	return 0;
}

int tree_callback_print_hc(TreeNode *node, int n, int level, void *ctx) {
	printf("%c ", n == 0 ? node->v : node->v2);
	return 0;
}

int tree_callback_print_v(TreeNode *node, int n, int level, void *ctx) {
	int i;
	for (i = 0; i < level; ++i) {
		printf("\t");
	}
	if (n == 0)
		printf("(");
	printf("%d", n == 0 ? node->v : node->v2);
	if ((node->twins && n == 1) || !node->twins)
		printf(")");
	printf("\n");
	return 0;
}

int tree_callback_print_vc(TreeNode *node, int n, int level, void *ctx) {
	int i;
	for (i = 0; i < level; ++i) {
		printf("\t");
	}
	if (n == 0)
		printf("(");
	printf("%c", n == 0 ? node->v : node->v2);
	if ((node->twins && n == 1) || !node->twins)
		printf(")");
	printf("\n");
	return 0;
}

/* return stop or not */
int tree_walk(TreeNode *node, int level, Callback cb, void *ctx) {
	if (node->left && tree_walk(node->left, level + 1, cb, ctx))
		return 1;
	if (cb(node, 0, level, ctx))
		return 1;
	if (node->twins) {
		if (node->center && tree_walk(node->center, level + 1, cb, ctx))
			return 1;
		if (cb(node, 1, level, ctx))
			return 1;
	}
	if (node->right && tree_walk(node->right, level + 1, cb, ctx))
		return 1;
	return 0;
}

static void tree_node_init_single(TreeNode* node, T v, TreeNode* left,
		TreeNode* right) {
	//shrink left to Single Node
	node->v = v;
	node->twins = 0;
	node->v2 = 0;
	node->left = left;
	if (left)
		left->up = node;
	node->center = NULL;
	node->right = right;
	if (right)
		right->up = node;
}

static TreeNode* tree_node_alloc(TreeNode* up, T v, TreeNode* left,
		TreeNode* right) {
	// new upper Single Node
	TreeNode* node = (TreeNode*) calloc(sizeof(TreeNode), 1);
	tree_node_init_single(node, v, left, right);
	node->up = up;
	return node;
}

static TreeNode *tree_node_insert(TreeNode *node, T v, TreeNode *left,
		TreeNode *right);

static TreeNode *tree_node_split(TreeNode *node, T v1, T v2, T v3, //
		TreeNode *n1, TreeNode *n2, TreeNode *n3, TreeNode *n4) {
	TreeNode *left = node; //reuse node
	//shrink left to Single Node
	tree_node_init_single(left, v1, n1, n2);

	//Create new right Single Node
	TreeNode *right = tree_node_alloc(node->up, v3, n3, n4);

	TreeNode *up = node->up;
	if (up == NULL) { // new upper Single Node
		up = tree_node_alloc(NULL, v2, left, right);
		return up;
	} else { //upper node exist, escalate
		return tree_node_insert(up, v2, left, right);
	}
}

/* insert v to up:
 * if up is single, make it twins
 * if up is twins, split
 * return NULL if no new top tree node created;
 */
static TreeNode *tree_node_insert(TreeNode *node, T v, TreeNode *left,
		TreeNode *right) {
	if (!node->twins) { //Single node -> Twins
		node->twins = 1;
		if (v < node->v) {
			node->v2 = node->v;
			node->v = v;
			node->left = left;
			node->center = right;
		} else {
			node->v2 = v;
			node->center = left;
			node->right = right;
		}
		return NULL;
	} else { //twins, must have 3 child, split and escalate the middle one
		if (v < node->v) {
			node = tree_node_split(node, v, node->v, node->v2, left, right,
					node->center, node->right);
		} else if (v < node->v2) {
			node = tree_node_split(node, node->v, v, node->v2, node->left, left,
					right, node->right);
		} else {
			node = tree_node_split(node, node->v, node->v2, v, node->left,
					node->center, left, right);
		}
		return node;
	}
}

static void tree_node_check(TreeNode* node) {
	assert(node);
	assert(
			(node->left && node->right) || (node->left == NULL && node->right == NULL));
	if (node->left)
		assert(node->left->up == node);
	if (node->center)
		assert(node->center->up == node);
	if (node->right)
		assert(node->right->up == node);
}

/* NULL: v exists, no action */
static TreeNode *tree_search_leaf_add(TreeNode *node, T v) {
	tree_node_check(node);

	if (v == node->v || (node->twins && node->v2 == v))
		return NULL;

	if (node->left) { // has children, 2 or 3
		if (v < node->v)  //less than v
			return tree_search_leaf_add(node->left, v);
		if (node->twins && v < node->v2) //twins node
			return tree_search_leaf_add(node->center, v);
		else
			return tree_search_leaf_add(node->right, v);
	}
	return node;
}

/* TreeNode grow strategy:
 * 	1. Grow up from leaf!
 * 	2. Single -> Twins, so each twins node must have 3 children
 * 	3. Twins -> 3 Single, so each new parent must have 2 children
 * 	4. left and right must exist!
 * 	5. if twins, center must exist!
 */
TreeNode *tree_add(TreeNode *tree, T v) {

	if (tree == NULL)
		return tree_node_alloc(NULL, v, NULL, NULL);

	tree_node_check(tree);

	TreeNode *leaf = tree_search_leaf_add(tree, v);
	if (!leaf)  //already exits on tree
		return tree;

	TreeNode *node = tree_node_insert(leaf, v, NULL, NULL);
	return node ? node : tree;
}

void tree_test_number(int n, int random) {
	TreeNode* t;
	int i;
	T last = 0;
	T v;
	for (i = 1; i <= n; ++i) {
		v = random ? ((double) rand() * n) / RAND_MAX : i;
		printf("%d ", v);
		t = tree_add(t, v);
//		tree_walk(t, 0, tree_callback_print_v, 0);
//		printf("===================\n");
	}
	printf("\n");
	tree_walk(t, 0, tree_callback_verify, &last);
	tree_walk(t, 0, tree_callback_print_h, NULL);
	printf("\n");
	tree_walk(t, 0, tree_callback_print_v, 0);
}

void tree_test_chars() {
	TreeNode *t = NULL;
	int i;
	T last = 0;
	char c;
// http://blog.csdn.net/yang_yulei/article/details/26066409
	char* str = "EARCHXMPL";
	for (i = 0; i < strlen(str); ++i) {
		c = str[i]; //random();
		printf("%c ", c);
		t = tree_add(t, c);
	}
	printf("\n");

	tree_walk(t, 0, tree_callback_verify, &last);
	tree_walk(t, 0, tree_callback_print_hc, NULL);
	printf("\n");
	tree_walk(t, 0, tree_callback_print_vc, NULL);
}

int main(int argc, char **argv) {
	tree_test_chars();
	tree_test_number(20, 1);
	return EXIT_SUCCESS;
}