\u6bcf\u4e2a\u8282\u70b9\u6709\u96f6\u4e2a\u6216\u591a\u4e2a\u5b50\u8282\u70b9<\/p>\n<\/li>\n
\u6ca1\u6709\u7236\u8282\u70b9\u7684\u8282\u70b9\u79f0\u4e3a\u6839\u8282\u70b9<\/p>\n<\/li>\n
\u6ca1\u6709\u5b50\u8282\u70b9\u7684\u8282\u70b9\u79f0\u4e3a\u53f6\u8282\u70b9<\/p>\n<\/li>\n
\u6bcf\u4e2a\u975e\u6839\u8282\u70b9\u6709\u4e14\u53ea\u6709\u4e00\u4e2a\u7236\u8282\u70b9<\/p>\n<\/li>\n<\/ul>\n
\u6811\u7684\u672f\u8bed<\/p>\n
- \n
- \n
\u5ea6(Degree): \u8282\u70b9\u7684\u5b50\u6811\u4e2a\u6570<\/p>\n<\/li>\n
- \n
\u6df1\u5ea6(Depth): \u6839\u8282\u70b9\u5230\u8be5\u8282\u70b9\u7684\u8def\u5f84\u957f\u5ea6<\/p>\n<\/li>\n
- \n
\u9ad8\u5ea6(Height): \u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u6700\u957f\u8def\u5f84<\/p>\n<\/li>\n
- \n
\u5c42\u6b21(Level): \u6839\u8282\u70b9\u4e3a\u7b2c1\u5c42，\u4f9d\u6b21\u9012\u589e<\/p>\n<\/li>\n<\/ul>\n
\u666e\u901a\u6811\u7684Python\u5b9e\u73b0<\/p>\n
class TreeNode:
\n """\u6811\u8282\u70b9\u7c7b"""
\n def __init__(self, data):
\n self.data = data # \u8282\u70b9\u6570\u636e
\n self.children = [] # \u5b50\u8282\u70b9\u5217\u8868<\/p>\ndef add_child(self, child_node):
\n """\u6dfb\u52a0\u5b50\u8282\u70b9"""
\n self.children.append(child_node)<\/p>\ndef __str__(self, level=0):
\n """\u53ef\u89c6\u5316\u6811\u7ed3\u6784"""
\n ret = " " * level + str(self.data) + "\\\\n" #\u5148\u5199\u597d\u6839\u8282\u70b9\u6570\u636e
\n for child in self.children:
\n ret += child.__str__(level + 1)
\n return ret<\/p>\n# \u521b\u5efa\u6811
\nroot = TreeNode("\u516c\u53f8")
\nsales = TreeNode("\u9500\u552e\u90e8")
\nsales.add_child(TreeNode("\u9500\u552e\u4e00\u7ec4"))
\nsales.add_child(TreeNode("\u9500\u552e\u4e8c\u7ec4"))<\/p>\ntech = TreeNode("\u6280\u672f\u90e8")
\ntech.add_child(TreeNode("\u5f00\u53d1\u7ec4"))
\ntech.add_child(TreeNode("\u6d4b\u8bd5\u7ec4"))<\/p>\nroot.add_child(sales)
\nroot.add_child(tech)<\/p>\nprint(root)
\n# \u8f93\u51fa:
\n# \u516c\u53f8
\n# \u9500\u552e\u90e8
\n# \u9500\u552e\u4e00\u7ec4
\n# \u9500\u552e\u4e8c\u7ec4
\n# \u6280\u672f\u90e8
\n# \u5f00\u53d1\u7ec4
\n# \u6d4b\u8bd5\u7ec4<\/p>\n\u4e8c\u3001\u4e8c\u53c9\u6811（Binary Tree）<\/h2>\n
\u57fa\u672c\u6982\u5ff5<\/p>\n
\u4e8c\u53c9\u6811\u662f\u6bcf\u4e2a\u8282\u70b9\u6700\u591a\u6709\u4e24\u4e2a\u5b50\u8282\u70b9\u7684\u6811\u7ed3\u6784，\u79f0\u4e3a：<\/p>\n
- \n
- \n
\u5de6\u5b50\u8282\u70b9<\/p>\n<\/li>\n
- \n
\u53f3\u5b50\u8282\u70b9<\/p>\n<\/li>\n<\/ul>\n
\u4e8c\u53c9\u6811\u7684\u6027\u8d28<\/p>\n
- \n
\u7b2c
![\"i\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/07\/20250730023218-6889843221b65.png\")
\u5c42\u6700\u591a\u6709![\"2^{i-1}\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/07\/20250730023218-68898432408f9.png\")
\u4e2a\u8282\u70b9<\/p>\n<\/li>\n - \n
\u6df1\u5ea6\u4e3a
![\"k\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/07\/20250730023218-68898432487f2.png\")
\u7684\u4e8c\u53c9\u6811\u6700\u591a\u6709![\"2^k-1\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/07\/20250730023218-688984324fe8f.png\")
\u4e2a\u8282\u70b9<\/p>\n<\/li>\n - \n
\u5bf9\u4e8e\u4efb\u4f55\u975e\u7a7a\u4e8c\u53c9\u6811，
![\"n_0](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/07\/20250730023218-6889843259536.png\")
（![\"n_0\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/07\/20250730023218-6889843264c8a.png\")
：\u53f6\u8282\u70b9\u6570，![\"n_2\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/07\/20250730023218-688984326ce88.png\")
：\u5ea6\u4e3a2\u7684\u8282\u70b9\u6570）<\/p>\n<\/li>\n\u00a0Python\u5b9e\u73b0\u4e8c\u53c9\u6811<\/p>\n
class BinaryTreeNode:
\n """\u4e8c\u53c9\u6811\u8282\u70b9\u7c7b"""<\/p>\ndef __init__(self, value):
\n self.value = value # \u8282\u70b9\u503c
\n self.left = None # \u5de6\u5b50\u8282\u70b9
\n self.right = None # \u53f3\u5b50\u8282\u70b9<\/p>\ndef __str__(self):
\n """\u8fd4\u56de\u6a2a\u8eba\u5b57\u7b26\u753b\u5b57\u7b26\u4e32"""
\n lines = []
\n self._build_lines(self, "", True, lines)
\n return "\\\\n".join(lines)<\/p>\n# \u5185\u90e8\u9012\u5f52，\u628a\u6bcf\u4e00\u884c append \u5230 lines \u5217\u8868
\n def _build_lines(self, node, prefix, is_left, lines):
\n if node is None:
\n return
\n self._build_lines(node.right,
\n prefix + ("\u2502 " if is_left else " "), False, lines)
\n connector = "\u2514\u2500\u2500 " if is_left else "\u250c\u2500\u2500 "
\n if prefix == "":
\n connector = ""
\n lines.append(prefix + connector + str(node.value))
\n self._build_lines(node.left,
\n prefix + (" " if is_left else "\u2502 "), True, lines)<\/p>\n# \u521b\u5efa\u4e8c\u53c9\u6811
\nroot = BinaryTreeNode(1)
\nroot.left = BinaryTreeNode(2)
\nroot.right = BinaryTreeNode(3)
\nroot.left.left = BinaryTreeNode(4)
\nroot.left.right = BinaryTreeNode(5)<\/p>\nprint(root)
\n# \u8f93\u51fa:
\n# \u2502 \u250c\u2500\u2500 3
\n# 1
\n# \u2502 \u250c\u2500\u2500 5
\n# \u2514\u2500\u2500 2
\n# \u2514\u2500\u2500 4<\/p>\n\u4e8c\u53c9\u6811\u904d\u5386\u65b9\u5f0f<\/p>\n
- \n
- \u524d\u5e8f\u904d\u5386(Pre-order): \u6839 -> \u5de6 -> \u53f3<\/li>\n
- \u4e2d\u5e8f\u904d\u5386(In-order): \u5de6 -> \u6839 -> \u53f3<\/li>\n
- \u540e\u5e8f\u904d\u5386(Post-order): \u5de6 -> \u53f3 -> \u6839<\/li>\n
- \u5c42\u5e8f\u904d\u5386(Level-order): \u6309\u5c42\u6b21\u4ece\u4e0a\u5230\u4e0b\u3001\u4ece\u5de6\u5230\u53f3<\/li>\n<\/ul>\n
def preorder_traversal(node):
\n """\u524d\u5e8f\u904d\u5386"""
\n if node:
\n print(node.value, end=" ")
\n preorder_traversal(node.left)
\n preorder_traversal(node.right)<\/p>\ndef inorder_traversal(node):
\n """\u4e2d\u5e8f\u904d\u5386"""
\n if node:
\n inorder_traversal(node.left)
\n print(node.value, end=" ")
\n inorder_traversal(node.right)<\/p>\ndef postorder_traversal(node):
\n """\u540e\u5e8f\u904d\u5386"""
\n if node:
\n postorder_traversal(node.left)
\n postorder_traversal(node.right)
\n print(node.value, end=" ")<\/p>\ndef levelorder_traversal(node):
\n """\u5c42\u5e8f\u904d\u5386"""
\n if not node:
\n return
\n queue = [node]
\n while queue:
\n current = queue.pop(0)
\n print(current.value, end=" ")
\n if current.left:
\n queue.append(current.left)
\n if current.right:
\n queue.append(current.right)<\/p>\n# \u6d4b\u8bd5\u904d\u5386
\nprint("\\\\n\u524d\u5e8f\u904d\u5386:", end=" ")
\npreorder_traversal(root) # \u8f93\u51fa: 1 2 4 5 3<\/p>\nprint("\\\\n\u4e2d\u5e8f\u904d\u5386:", end=" ")
\ninorder_traversal(root) # \u8f93\u51fa: 4 2 5 1 3<\/p>\nprint("\\\\n\u540e\u5e8f\u904d\u5386:", end=" ")
\npostorder_traversal(root) # \u8f93\u51fa: 4 5 2 3 1<\/p>\nprint("\\\\n\u5c42\u5e8f\u904d\u5386:", end=" ")
\nlevelorder_traversal(root) # \u8f93\u51fa: 1 2 3 4 5<\/p>\n\u4e09\u3001\u4e8c\u53c9\u641c\u7d22\u6811（BST-Binary Search Tree）<\/h2>\n
\u4e8c\u53c9\u641c\u7d22\u6811\u662f\u4e00\u79cd\u7279\u6b8a\u7684\u4e8c\u53c9\u6811，\u6ee1\u8db3：<\/p>\n
- \n
- \u5de6\u5b50\u6811\u6240\u6709\u8282\u70b9\u7684\u503c < \u6839\u8282\u70b9\u7684\u503c<\/li>\n
- \u53f3\u5b50\u6811\u6240\u6709\u8282\u70b9\u7684\u503c > \u6839\u8282\u70b9\u7684\u503c<\/li>\n
- \u5de6\u53f3\u5b50\u6811\u4e5f\u5206\u522b\u662fBST<\/li>\n<\/ul>\n
BST\u7684\u6027\u8d28<\/p>\n
- \n
- \u4e2d\u5e8f\u904d\u5386BST\u4f1a\u5f97\u5230\u4e00\u4e2a\u5347\u5e8f\u5e8f\u5217<\/li>\n
- \u67e5\u627e\u3001\u63d2\u5165\u3001\u5220\u9664\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3aO(log n)，\u6700\u574fO(n)（\u9000\u5316\u4e3a\u94fe\u8868\u65f6）<\/li>\n<\/ul>\n
Python\u5b9e\u73b0BST<\/p>\n
class BSTNode:
\n """\u4e8c\u53c9\u641c\u7d22\u6811\u8282\u70b9\u7c7b"""<\/p>\ndef __init__(self, key):
\n self.key = key # \u8282\u70b9\u952e\u503c
\n self.left = None # \u5de6\u5b50\u8282\u70b9
\n self.right = None # \u53f3\u5b50\u8282\u70b9
\n self.parent = None # \u7236\u8282\u70b9（\u53ef\u9009）<\/p>\ndef __str__(self):
\n """\u53ef\u89c6\u5316BST\u7ed3\u6784"""
\n return f"{self.key}(L:{self.left}, R:{self.right})"<\/p>\nclass BinarySearchTree:
\n """\u4e8c\u53c9\u641c\u7d22\u6811\u7c7b"""<\/p>\ndef __init__(self):
\n self.root = None<\/p>\ndef insert(self, key):
\n """\u63d2\u5165\u8282\u70b9"""
\n if self.root is None:
\n self.root = BSTNode(key)
\n else:
\n self._insert_recursive(self.root, key)<\/p>\ndef _insert_recursive(self, node, key):
\n """\u9012\u5f52\u63d2\u5165\u8f85\u52a9\u51fd\u6570"""
\n if key < node.key:
\n if node.left is None:
\n node.left = BSTNode(key)
\n node.left.parent = node
\n else:
\n self._insert_recursive(node.left, key)
\n elif key > node.key:
\n if node.right is None:
\n node.right = BSTNode(key)
\n node.right.parent = node
\n else:
\n self._insert_recursive(node.right, key)
\n # \u5982\u679ckey == node.key，\u53ef\u4ee5\u9009\u62e9\u5ffd\u7565\u6216\u5904\u7406\u91cd\u590d\u503c
\n elif key == node.key:
\n # \u7b56\u7565 1：\u76f4\u63a5\u5ffd\u7565（\u6700\u7b80\u5355）
\n return<\/p>\n# \u7b56\u7565 2：\u8ba1\u6570\u6cd5（\u7ed9\u8282\u70b9\u52a0\u4e00\u4e2a count \u5c5e\u6027）
\n # node.count += 1<\/p>\ndef search(self, key):
\n """\u67e5\u627e\u8282\u70b9"""
\n return self._search_recursive(self.root, key)<\/p>\ndef _search_recursive(self, node, key):
\n """\u9012\u5f52\u67e5\u627e\u8f85\u52a9\u51fd\u6570"""
\n if node is None or node.key == key:
\n return node
\n if key < node.key:
\n return self._search_recursive(node.left, key)
\n return self._search_recursive(node.right, key)<\/p>\ndef delete(self, key):
\n """\u5220\u9664\u8282\u70b9"""
\n self.root = self._delete_recursive(self.root, key)<\/p>\ndef _delete_recursive(self, node, key):
\n """\u9012\u5f52\u5220\u9664\u8f85\u52a9\u51fd\u6570"""
\n if node is None:
\n return node<\/p>\nif key < node.key:
\n node.left = self._delete_recursive(node.left, key)
\n elif key > node.key:
\n node.right = self._delete_recursive(node.right, key)
\n else:
\n # \u627e\u5230\u8981\u5220\u9664\u7684\u8282\u70b9
\n if node.left is None: #\u56ca\u62ec\u4e86\u5de6\u53f3\u5b50\u6811\u5747\u4e3a\u7a7a\u7684\u60c5\u51b5\u3002
\n return node.right
\n elif node.right is None:
\n return node.left
\n else:
\n # \u6709\u4e24\u4e2a\u5b50\u8282\u70b9，\u627e\u5230\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u8282\u70b9\u66ff\u4ee3，\u56e0\u4e3a\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u503c\u4e00\u5b9a\u6bd4\u5de6\u5b50\u6811\u5927，\u4f46\u53c8\u6bd4\u53f3\u5b50\u6811\u5c0f
\n successor = self._find_min(node.right)
\n node.key = successor.key
\n node.right = self._delete_recursive(node.right, successor.key)
\n return node<\/p>\ndef _find_min(self, node):
\n """\u627e\u5230\u5b50\u6811\u4e2d\u7684\u6700\u5c0f\u8282\u70b9"""
\n current = node
\n while current.left is not None:
\n current = current.left
\n return current<\/p>\ndef inorder_traversal(self):
\n """\u4e2d\u5e8f\u904d\u5386（\u8fd4\u56de\u6709\u5e8f\u5e8f\u5217）"""
\n result = []
\n self._inorder_recursive(self.root, result)
\n return result<\/p>\ndef _inorder_recursive(self, node, result):
\n """\u4e2d\u5e8f\u904d\u5386\u8f85\u52a9\u51fd\u6570"""
\n if node:
\n self._inorder_recursive(node.left, result)
\n result.append(node.key)
\n self._inorder_recursive(node.right, result)<\/p>\n# \u6d4b\u8bd5BST
\nbst = BinarySearchTree()
\nkeys = [50, 30, 70, 20, 40, 60, 80]<\/p>\nfor key in keys:
\n bst.insert(key)<\/p>\nprint("\u4e2d\u5e8f\u904d\u5386:", bst.inorder_traversal()) # \u8f93\u51fa: [20, 30, 40, 50, 60, 70, 80]<\/p>\n
# \u67e5\u627e\u6d4b\u8bd5
\nprint("\u67e5\u627e40:", bst.search(40) is not None) # \u8f93\u51fa: True
\nprint("\u67e5\u627e45:", bst.search(45) is not None) # \u8f93\u51fa: False<\/p>\n# \u5220\u9664\u6d4b\u8bd5
\nbst.delete(30)
\nprint("\u5220\u966430\u540e\u4e2d\u5e8f\u904d\u5386:", bst.inorder_traversal()) # \u8f93\u51fa: [20, 40, 50, 60, 70, 80]<\/p>\n","protected":false},"excerpt":{"rendered":"\u6587\u7ae0\u6d4f\u89c8\u9605\u8bfb243\u6b21\u3002\u6811\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u7531\u8282\u70b9\u548c\u8fb9\u7ec4\u6210\uff0c\u5177\u6709\u4ee5\u4e0b\u7279\u70b9\uff1a\u6bcf\u4e2a\u8282\u70b9\u6709\u96f6\u4e2a\u6216\u591a\u4e2a\u5b50\u8282\u70b9\u6ca1\u6709\u7236\u8282\u70b9\u7684\u8282\u70b9\u79f0\u4e3a\u6839\u8282\u70b9\u6ca1\u6709\u5b50\u8282\u70b9\u7684\u8282\u70b9\u79f0\u4e3a\u53f6\u8282\u70b9\u6bcf\u4e2a\u975e\u6839\u8282\u70b9\u6709\u4e14\u53ea\u6709\u4e00\u4e2a\u7236\u8282\u70b9\u5ea6(Degree): \u8282\u70b9\u7684\u5b50\u6811\u4e2a\u6570\u6df1\u5ea6(Depth): \u6839\u8282\u70b9\u5230\u8be5\u8282\u70b9\u7684\u8def\u5f84\u957f\u5ea6\u9ad8\u5ea6(Height): \u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u6700\u957f\u8def\u5f84\u5c42\u6b21(Level): \u6839\u8282\u70b9\u4e3a\u7b2c1\u5c42\uff0c\u4f9d\u6b21\u9012\u589e\u3002<\/p>\n","protected":false},"author":2,"featured_media":48035,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[81,4631,190],"topic":[],"class_list":{"0":"post-48042","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","6":"hentry","7":"category-server","8":"tag-python","10":"tag-190"},"yoast_head":"\n
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