Binary Search Trees: Structure, Traversals, and Deletion Algorithms

Fundamentals of Binary Search Trees

• Binary Search Tree (BST) Definition:

  • A Binary Search Tree is a specialized category of binary tree constructed to optimize key searching and node insertion operations compared to standard binary trees.

  • Unlike general binary trees that lack structural key-ordering rules, a BST imposes explicit relative ordering constraints between parent nodes and their left and right subtrees.

• Generic Tree Operations:

  • Deletion and search routines can be implemented generically to support nodes containing any data type (e.g., numbers, characters, or strings).

  • When node values are non-numeric (such as letters or strings), ordering and replacement decisions follow in-order successor or in-order predecessor relationships dictated by character collating sequences or alphabetical order.

Binary Search Tree Node Deletion Mechanics

• Structural Invariant Rule:

  • Any node deletion operation must preserve all valid binary search tree properties across the entire tree upon completion.

• Single-Child Node Replacement:

  • When a target node for deletion possesses only one child, the node is replaced directly by its child.

• Two-Child / Internal Node Replacement Strategies:

  • In-Order Predecessor Strategy:

  • Definition: The in-order predecessor is the node that immediately precedes the target removal node when performing an in-order traversal.

  • Identification Procedure: Navigate to the target node's left subtree and select the node containing the largest value in that left subtree.

  • Example: To remove node $4$, locate its left subtree; performing an in-order traversal identifies node $3$ as the predecessor. Node $4$ is replaced by node $3$.

  • In-Order Successor Strategy:

  • Definition: The in-order successor is the node that immediately follows the target removal node when performing an in-order traversal.

  • Identification Procedure: Navigate to the target node's right subtree and select the node containing the smallest value in that right subtree.

  • Example: To remove node $4$, locate its right subtree; performing an in-order traversal identifies node $5$ as the immediate successor. Node $4$ is replaced by node 5$.\n\n# Step-by-Step BST Construction, Insertion, and Deletion Example\n\n* Initial Setup:\n * Tree initialization begins with an empty binary search tree.\n * Initial root node set as 44.\n * Rendered layout initial root node is 24$, with left child node$17$and right child node$88$.

• Insertion Operations:

  • Inserting Node $76$:

  • Step 1: Compare $76$ to root node $24$. Since $76 > 24$, navigate to the right child node ($88$).

  • Step 2: Compare $76$ to node $88$. Since $76 < 88$, navigate to the left child position of $88$ and place node $76$.

  • Inserting Node $68$:

  • Step 1: Compare $68$ to root node $24$. Since $68 > 24$, navigate right toward node $88$.

  • Step 2: Compare $68$ to node $88$. Since $68 < 88$, navigate left toward node 76$.\n * Step 3: Compare 68$to node$76$. Since$68 < 76$, navigate left to insert node$68 in the terminal leaf position.\n* Deletion Operations:\n * **Deleting Node 32$(or$33)**:\n * Step 1: Search and locate target node 32$/$33 in the tree.\n * Step 2: Determine child count (evaluating node 38).\n * **Deleting Node 8**:\n * Step 1: Locate node 8 within the tree.\n * Step 2: Verify presence of child nodes (node 8 possesses children).\n * Step 3: Choose replacement strategy using available subtree paths.\n * Step 4: Perform node replacement using node 29$.

  • Deleting Root Node $44$:

  • Evaluate left subtree vs. right subtree replacement paths.

  • When choosing the right subtree path, extract the smallest node value in that right subtree to maintain valid BST ordering.

Test Statistics

• Summary of Class Assessment Results:

  • The class average score on the test was almost calculated/completed.