Binary-Search-Tree-Property

Assume all the numbers in the following tree are unique integers.

Choose the correct statements about the binary search tree.

Question

Binary-Search-Tree-Property

Assume all the numbers in the following tree are unique integers.

Choose the correct statements about the binary search tree.

BlackTom AI · Accepted Answer

BST invariant: for every node X, all keys in X's left subtree < X.key < all keys in X's right subtree. 1) Relevant knowledge points and concepts - BST ordering/invariant (left < node < right). - Inorder predecessor/successor rules: - Successor of a node with a right subtree = the minimum (leftmost) node in that right subtree. - If no right subtree, successor = first ancestor for which the node is in that ancestor's left subtree. - Predecessor similarly uses left subtree max or ancestor where node is in right subtree. 2) Step-by-step reasoning for bounds on B - 56 is the root. Everything in its left subtree (including A, 18, 24, B, C, etc.) must be < 56 → every such node < 56. - Node 18 is in A's left subtree, so 18 < A. - 24 is the right child of 18, hence 18 < 24. Because 24 is still in A's left subtree, 24 < A. So we have 18 < 24 < A < 56. - B is the right child (or in the right subtree) of A, so B > A. Combining: A > 24 > 18 ⇒ B > A > 24 > 18. - Also, because B is in the left subtree of 56, B < 56. - Therefore 18 < B < 56 (strictly). This confirms: "The value of B must be between 18 and 56, exclusively." 3) Reasoning for successor of B - The diagram shows B has no right subtree (only a single left child C is shown), so the successor rule for a node with no right subtree applies. - Walk upward from B to find the first ancestor for which B is in that ancestor's left subtree: - B is in A's right subtree, so A is not such an ancestor. - A is the left child of 56, so B is in the left subtree of 56. Thus 56 is the first ancestor where B is in that ancestor's left subtree. - Hence the inorder successor of B is 56. This confirms: "The successor of B is 56." 4) Why the correct answers are correct - The inequality 18 < B < 56 follows directly from the local subtree inequalities shown above. - With no right child of B, the successor must be the ancestor found by the ancestor-rule; that ancestor is 56. 5) Why the other options are incorrect (brief) - "The predecessor of the root is A": incorrect. Predecessor of 56 is the maximum in its left subtree, which is B (largest node < 56), not A. - "The tree is a complete binary tree": incorrect. Leaves at the lowest level are not all present/filled left-to-right, so it's not complete. - "The successor of 24 is B": incorrect. 24 has no right subtree, so its successor is the first ancestor where 24 is in the ancestor's left subtree; that ancestor is A (not B). 6) Conclusion Both selected statements — "The value of B must be between 18 and 56, exclusively" and "The successor of B is 56" — are correct, matching the provided answers.

Binary-Search-Tree-Property Assume all the numbers in the following tree are unique integers. Choose the correct statements about the binary search tree. 多项选择题

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