105/106. Construct Binary Tree from Inorder and Postorder(Preorder) Traversal

根據前序，中序或者後序，中序重構樹。這裡僅以後序+中序進行分析。文末給出前序的類似程式碼，原理一致。

Notes

• 遞迴的難點在於遞迴函式的正確定義
• 理解定義再理解程式碼非常簡單，尤其是二叉樹，對於左子樹的遞迴，正確的理解是，左子樹任務已經完成了，我需要做什麼
• 設計遞迴函式，可以從基元考慮（沒有子節點，邊界節點）和普遍情況考慮（即左子節點全部完成，右子節點全部完成），前者是自底向上，後者是自頂向下的思維。

Version 1

回到重構二叉樹的問題，這其實是個常見的簡單問題，重點就是把座標細節搞定。

• 新建當前節點
• 找到左子樹範圍，重構左子樹
• 找到右子樹範圍，重構右子樹

# version 1
# 遞迴構造
# 時間，每層時間和N，深度最多N，一般情況是logN，所以最差N^2，一般情況NlogN
# 額外空間，無
# Runtime: 356 ms, faster than 10.68% of Python online submissions for Construct Binary Tree from Inorder and Postorder Traversal.
# Memory Usage: 18.7 MB, less than 60.97% of Python online submissions for Construct Binary Tree from Inorder and Postorder Traversal.
class Solution(object):
    def buildTree(self, inorder, postorder):
        def rebuild(pl, ph, il, ih):
            if il >= ih: return None
            root = TreeNode(val = postorder[ph-1])
            # search in indorder
            for idx in range(il, ih):
                if inorder[idx] == root.val: break
            mid = ph-1-(ih-1-idx)
            root.right = rebuild(mid, ph-1, idx+1, ih)
            root.left = rebuild(pl, mid, il, idx)
            return root
        return rebuild(0, len(inorder), 0, len(inorder))

Version 2

利用map來保證$O(N)$的時間效率。你永遠都可以考慮優先用空間換時間。

# version 2
# 空間換時間
# Runtime: 52 ms, faster than 66.41% of Python online submissions for Construct Binary Tree from Inorder and Postorder Traversal.
# Memory Usage: 19.4 MB, less than 53.98% of Python online submissions for Construct Binary Tree from Inorder and Postorder Traversal.
class Solution(object):
    def buildTree(self, inorder, postorder):
        num2idx = dict(zip(inorder, range(len(inorder))))
        def rebuild(pl, ph, il, ih):
            if il >= ih: return None
            root = TreeNode(val = postorder[ph-1])
            # search in indorder
            idx = num2idx[root.val]
            mid = ph-1-(ih-1-idx)
            root.right = rebuild(mid, ph-1, idx+1, ih)
            root.left = rebuild(pl, mid, il, idx)
            return root
        return rebuild(0, len(inorder), 0, len(inorder))

Version 3

一種嚴格的$O(N)$演算法。非常優雅的實現。

理解該演算法並不容易，要點就是要記住遞迴函式的定義，然後相信它。它不再是單獨重構某一邊的子樹，而是定義為：給定stop，高速你在inoreder找到什麼停止，函式負責把我的整顆樹從陣列裡重構出來。

所有節點pop一次，所以複雜度$O(N)$.

話又說回來，遞迴演算法，定義是最重要的，思考的關鍵是相信定義。可是，這比較適合用來理解遞迴程式碼。設計還是很難，這道題的實現非常優雅。

# version  3 O（n）的演算法
# Runtime: 32 ms, faster than 99.61% of Python online submissions for Construct Binary Tree from Inorder and Postorder Traversal.
# Memory Usage: 17.7 MB, less than 94.95% of Python online submissions for Construct Binary Tree from Inorder and Postorder Traversal.
class Solution(object):
    def buildTree(self, inorder, postorder):
        # 告知stop，從二個陣列重構整顆樹，初始的stop為None
        def rebuild(stop):
            if  inorder[-1] != stop:
                # 新建當前節點
                root  = TreeNode(val=postorder.pop())
                # 重構右子樹，inoreder的stop為當前節點的值，找到則右子樹重構完畢
                root.right = rebuild(root.val)
                # 注意右子樹重構完畢，意味著inorder中已經不存在左子樹的節點了
                # 那麼現在該幹什麼？當然是pop掉當前的節點了
                inorder.pop()
                # 重構左子樹，左子樹的stop值為本函式的stop相同，初始為None
                root.left = rebuild(stop)
                return root
        # 哨兵節點
        inorder = [None] + inorder
        return rebuild(None)

附，前序+中序的重構

# version 1
# 遞迴構造
# Runtime: 356 ms, faster than 10.68% of Python online submissions for Construct Binary Tree from Inorder and Postorder Traversal.
# Memory Usage: 18.7 MB, less than 60.97% of Python online submissions for Construct Binary Tree from Inorder and Postorder Traversal.
class Solution(object):
    def buildTree(self, inorder, postorder):
        def rebuild(pl, ph, il, ih):
            if il >= ih: return None
            root = TreeNode(val = postorder[ph-1])
            # search in indorder
            for idx in range(il, ih):
                if inorder[idx] == root.val: break
            mid = ph-1-(ih-1-idx)
            root.right = rebuild(mid, ph-1, idx+1, ih)
            root.left = rebuild(pl, mid, il, idx)
            return root
        return rebuild(0, len(inorder), 0, len(inorder))
    
# version 2
# 空間換時間
# Runtime: 52 ms, faster than 66.41% of Python online submissions for Construct Binary Tree from Inorder and Postorder Traversal.
# Memory Usage: 19.4 MB, less than 53.98% of Python online submissions for Construct Binary Tree from Inorder and Postorder Traversal.
class Solution(object):
    def buildTree(self, inorder, postorder):
        num2idx = dict(zip(inorder, range(len(inorder))))
        def rebuild(pl, ph, il, ih):
            if il >= ih: return None
            root = TreeNode(val = postorder[ph-1])
            # search in indorder
            idx = num2idx[root.val]
            mid = ph-1-(ih-1-idx)
            root.right = rebuild(mid, ph-1, idx+1, ih)
            root.left = rebuild(pl, mid, il, idx)
            return root
        return rebuild(0, len(inorder), 0, len(inorder))


# version  3 O（n）的演算法
# Runtime: 32 ms, faster than 99.61% of Python online submissions for Construct Binary Tree from Inorder and Postorder Traversal.
# Memory Usage: 17.7 MB, less than 94.95% of Python online submissions for Construct Binary Tree from Inorder and Postorder Traversal.
class Solution(object):
    def buildTree(self, inorder, postorder):
        # 告知stop，從二個陣列重構整顆樹，初始的stop為None
        def rebuild(stop):
            if  inorder[-1] != stop:
                # 新建當前節點
                root  = TreeNode(val=postorder.pop())
                # 重構右子樹，inoreder的stop為當前節點的值，找到則右子樹重構完畢
                root.right = rebuild(root.val)
                # 注意右子樹重構完畢，意味著inorder中已經不存在左子樹的節點了
                # 那麼現在該幹什麼？當然是pop掉當前的節點了
                inorder.pop()
                # 重構左子樹，左子樹的stop值為本函式的stop相同，初始為None
                root.left = rebuild(stop)
                return root
        # 哨兵節點
        inorder = [None] + inorder
        return rebuild(None)

Ref

• https://leetcode.com/problems/construct-binary-tree-from-preorder-and-inorder-traversal/discuss/34543/Simple-O(n)-without-map