BST property and invariant in Data Structures Theory - Step-by-Step Execution

Concept Flow - BST property and invariant

Start at root node

↓

Check left subtree

↓

All left nodes < current node?

No→BST property violated

Yes↓

Check right subtree

↓

All right nodes > current node?

No→BST property violated

Yes↓

Repeat for all nodes

↓

If all checks pass

→BST property holds

The BST property means every node's left subtree has smaller values and right subtree has larger values. This must hold for every node.

Execution Sample

Data Structures Theory

Node(10)
 /      \
5       15
/ \     / \
2  7   12  20

A BST where each node's left children are smaller and right children are larger, maintaining the BST property.

Analysis Table

Step	Node Checked	Left Subtree Values	Right Subtree Values	BST Property Holds?
1	10	[5, 2, 7]	[15, 12, 20]	Yes, all left < 10 and right > 10
2	5	[2]	[7]	Yes, all left < 5 and right > 5
3	15	[12]	[20]	Yes, all left < 15 and right > 15
4	2	[]	[]	Yes, no children
5	7	[]	[]	Yes, no children
6	12	[]	[]	Yes, no children
7	20	[]	[]	Yes, no children
8	All nodes checked	-	-	BST property holds for entire tree

💡 All nodes satisfy the BST property, so the tree is a valid BST.

State Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 8
Current Node	root (10)	5	15	20	All nodes checked
Left Subtree Values	N/A	[2]	[12]	[]	N/A
Right Subtree Values	N/A	[7]	[20]	[]	N/A
BST Property Holds	Unknown	Yes	Yes	Yes	Yes

Key Insights - 3 Insights

Why do we check all nodes, not just the root?

Because the BST property must hold at every node, not just the root. The execution_table shows checks at each node (steps 1-7).

What if a left subtree node is greater than the current node?

Then the BST property is violated. The flow would go to 'BST property violated' as shown in the concept_flow diagram.

Why do leaf nodes automatically satisfy the BST property?

Leaf nodes have no children, so no left or right subtree to violate the property, as shown in steps 4-7 in the execution_table.

Visual Quiz - 3 Questions

Look at the execution_table, what is the left subtree values checked at node 5?

A [2]

B [5, 2, 7]

C []

D [7]

💡 Hint

Check row with Step 2 in execution_table under 'Left Subtree Values'

At which step does the execution_table confirm all nodes have been checked?

A Step 1

B Step 7

C Step 8

D Step 3

💡 Hint

Look for the row mentioning 'All nodes checked' in execution_table

If a node's right subtree contains a value less than the node, what happens?

A BST property holds

B BST property violated

C Check continues to next node

D Tree becomes balanced

💡 Hint

Refer to concept_flow where right subtree check fails leading to violation

Concept Snapshot

BST Property:
- For each node, all left subtree nodes < node value
- All right subtree nodes > node value
- Must hold for every node
- Ensures efficient search, insert, delete
- Violation means tree is not a BST

Full Transcript

The BST property means every node's left subtree contains only smaller values and the right subtree only larger values. This rule applies to every node in the tree. We check each node's left and right subtree values to confirm this. Leaf nodes automatically satisfy the property as they have no children. If any node violates this rule, the tree is not a BST. The execution table shows step-by-step checks for each node, confirming the property holds throughout the tree.