Given n, generate all structurally unique BST's (binary search trees) that store values 1...n.

For example,
Given n = 3, your program should return all 5 unique BST's shown below.

1         3     3      2      1
\       /     /      / \      \
3     2     1      1   3      2
/     /       \                 \
2     1         2                 3

confused what 

"{1,#,2,3}"

 means? > read more on how binary tree is serialized on OJ.

OJ's Binary Tree Serialization:

The serialization of a binary tree follows a level order traversal, where '#' signifies a path terminator where no node exists below.

Here's an example:

1
/ \
2   3
/
4
\
5

The above binary tree is serialized as 

"{1,2,3,#,#,4,#,#,5}"

.

96. Unique Binary Search Trees 的扩展，96题只要算出所有不同BST的个数，这道题让把那些二叉树都表示出来。

用递归来解，划分左右子树，递归构造。

1. 根节点可以任取min ~ max (例如min = 1, max = n)，假如取定为i。
2. 则left subtree由min ~ i-1组成，假设可以有L种可能。right subtree由i+1 ~ max组成，假设有R种可能。生成所有可能的left/right subtree。
3 对于每个生成的left subtree/right subtree组合<T_left(p), T_right(q)>，p = 1...L，q = 1...R，添加上根节点i而组成一颗新树。

Python:

class TreeNode:
def __init__(self, x):
self.val = x
self.left = None
self.right = None

def __repr__(self):
if self:
serial = []
queue = [self]

while queue:
cur = queue[0]

if cur:
serial.append(cur.val)
queue.append(cur.left)
queue.append(cur.right)
else:
serial.append("#")

queue = queue[1:]

while serial[-1] == "#":
serial.pop()

return repr(serial)

else:
return None

class Solution:
# @return a list of tree node
def generateTrees(self, n):
return self.generateTreesRecu(1, n)

def generateTreesRecu(self, low, high):
result = []
if low > high:
result.append(None)
for i in xrange(low, high + 1):
left = self.generateTreesRecu(low, i - 1)
right = self.generateTreesRecu(i + 1, high)
for j in left:
for k in right:
cur = TreeNode(i)
cur.left = j
cur.right = k
result.append(cur)
return result

C++:

class Solution {
public:
vector<TreeNode *> generateTrees(int n) {
return genBST(1, n);
}

vector<TreeNode *> genBST(int min, int max) {
vector<TreeNode *> ret;
if(min>max) {
ret.push_back(NULL);
return ret;
}

for(int i=min; i<=max; i++) {
vector<TreeNode*> leftSubTree = genBST(min,i-1);
vector<TreeNode*> rightSubTree = genBST(i+1,max);
for(int j=0; j<leftSubTree.size(); j++) {
for(int k=0; k<rightSubTree.size(); k++) {
TreeNode *root = new TreeNode(i);
root->left = leftSubTree[j];
root->right = rightSubTree[k];
ret.push_back(root);
}
}
}

return ret;
}
};

 

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[LeetCode] 96. Unique Binary Search Trees 唯一二叉搜索树　

 

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