A quadtree is a tree data in which each internal node has exactly four children: topLeft, topRight, bottomLeft and bottomRight. Quad trees are often used to partition a two-dimensional space by recursively subdividing it into four quadrants or regions.

We want to store True/False information in our quad tree. The quad tree is used to represent a N * N

 boolean grid. For each node, it will be subdivided into four children nodes until the values in the region it represents are all the same. Each node has another two boolean attributes : isLeaf and val. isLeafis true if and only if the node is a leaf node. The val attribute for a leaf node contains the value of the region it represents.

For example, below are two quad trees A and B:

A:
+-------+-------+   T: true
|       |       |   F: false
|   T   |   T   |
|       |       |
+-------+-------+
|       |       |
|   F   |   F   |
|       |       |
+-------+-------+
topLeft: T
topRight: T
bottomLeft: F
bottomRight: F

B:               
+-------+---+---+
|       | F | F |
|   T   +---+---+
|       | T | T |
+-------+---+---+
|       |       |
|   T   |   F   |
|       |       |
+-------+-------+
topLeft: T
topRight:
     topLeft: F
     topRight: F
     bottomLeft: T
     bottomRight: T
bottomLeft: T
bottomRight: F

Your task is to implement a function that will take two quadtrees and return a quadtree that represents the logical OR (or union) of the two trees.

A:                 B:                 C (A or B):
+-------+-------+  +-------+---+---+  +-------+-------+
|       |       |  |       | F | F |  |       |       |
|   T   |   T   |  |   T   +---+---+  |   T   |   T   |
|       |       |  |       | T | T |  |       |       |
+-------+-------+  +-------+---+---+  +-------+-------+
|       |       |  |       |       |  |       |       |
|   F   |   F   |  |   T   |   F   |  |   T   |   F   |
|       |       |  |       |       |  |       |       |
+-------+-------+  +-------+-------+  +-------+-------+

Note:

1. Both A and B represent grids of size N * N.
2. N is guaranteed to be a power of 2.
3. If you want to know more about the quad tree, you can refer to its wiki.
4. The logic OR operation is defined as this: "A or B" is true if A is true, or if B is true, or if both A and B are true.

這道題又是一道四叉樹的題，說是給了我們兩個四叉樹，然後讓我們將二棵樹相交形成了一棵四叉樹，相交的機制採用的是或，即每個自區域相‘或’，題目中給的例子很好的說明了一些相‘或’的原則，比如我們看A和B中的右上結點，我們發現A樹的右上結點已經是一個值為true的葉結點，而B的右上結點還是一個子樹，那麼此時不論子樹裡有啥內容，我們相交後的樹的右上結點應該跟A樹的右上結點保持一致，假如A樹的右上結點值是false的話，相‘或’起不到任何作用，那麼相交後的樹的右上結點應該跟B樹的右上結點保持一致。那麼我們可以歸納出，只有某一個結點是葉結點了，我們看其值，如果是true，則相交後的結點和此結點保持一致，否則跟另一個結點保持一致。比較麻煩的情況是當兩個結點都不是葉結點的情況，此時我們需要對相對應的四個子結點分別呼叫遞迴函式，呼叫之後還需要進行進一步處理，因為一旦四個子結點的值相同，且都是葉結點的話，那麼此時應該合併為一個大的葉結點，參見程式碼如下：

class Solution {
public:
    Node* intersect(Node* quadTree1, Node* quadTree2) {
          if (quadTree1->isLeaf) return quadTree1->val ? quadTree1 : quadTree2;
          if (quadTree2->isLeaf) return quadTree2->val ? quadTree2 : quadTree1;
          Node *tl = intersect(quadTree1->topLeft, quadTree2->topLeft);
          Node *tr = intersect(quadTree1->topRight, quadTree2->topRight);
          Node *bl = intersect(quadTree1->bottomLeft, quadTree2->bottomLeft);
          Node *br = intersect(quadTree1->bottomRight, quadTree2->bottomRight);
          if (tl->val == tr->val && tl->val == bl->val && tl->val == br->val && tl->isLeaf && tr->isLeaf && bl->isLeaf && br->isLeaf) {
              return new Node(tl->val, true, NULL, NULL, NULL, NULL);
          } else {
              return new Node(false, false, tl, tr, bl, br);
          }
    }
};

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