Create a red-black tree by in serting following sequence of number8, 18, 5 · 1s, 17, ,d 1 25, 40 &80

To create a red-black tree by inserting the sequence of numbers 8, 18, 5, 1, 17, 25, 40, and 80, we will follow these steps:

1. Break Down the Problem

We need to insert the following numbers into a red-black tree in order, ensuring that all the properties of the red-black tree are maintained after each insertion:

• Insertion Sequence: 8, 18, 5, 1, 17, 25, 40, 80

2. Relevant Concepts

A red-black tree must satisfy the following properties:

1. Each node is either red or black.
2. The root is always black.
3. All leaves (NIL nodes) are black.
4. If a red node has children, then, both children must be black (no two red nodes can be adjacent).
5. Every path from a node to its descendant leaves has the same number of black nodes.

3. Analysis and Detail

We will perform the insertion one by one while applying necessary rotations and recoloring to maintain red-black properties.

• Insert 8: The tree contains only the root.

      8 (Black)
  

• Insert 18: 18 is greater than 8, so it becomes a red child:

      8 (Black)
         \
         18 (Red)
  

• Insert 5: 5 is less than 8, so it becomes a red child:

       8 (Black)
       /    \
     5 (Red) 18 (Red)
  

  Now, since we have two red nodes in a row (5 and 18), we need to perform a rotation. We may choose to color 8 red and both children black, since 8 was root:

       8 (Red)
       /    \
     5 (Black)  18 (Black)
  

• Insert 1: 1 is less than 5, so it is added as a red child of 5:

       8 (Red)
       /    \
     5 (Black)  18 (Black)
    /
  1 (Red)
  

  No violation yet.

• Insert 17: 17 is less than 18 but greater than 8, so it is a red child of 18:

       8 (Red)
       /    \
     5 (Black)  18 (Black)
    /         /
  1 (Red)  17 (Red)
  

  Now we have two red nodes in a row (17 and 18). We will perform a recoloring:

       8 (Red)
       /    \
     5 (Black)  18 (Red)
    /         /
  1 (Red)  17 (Black)
  

  Again, we have two reds on the same path. We perform a left rotation at 18 and recolor:

       8 (Black)
       /    \
     5 (Black)  17 (Black)
    /         /   \
  1 (Red)  18 (Red)
  

• Insert 25: 25 is greater than 18, so add it as a red child of 17:

       8 (Black)
       /    \
     5 (Black)  17 (Black)
    /         /   \
  1 (Red)  18 (Red) 25 (Red)
  

  Now both 18 and 25 are red. This can be solved by performing recoloring.

       8 (Black)
       /    \
     5 (Black)  17 (Red)
    /         /   \
  1 (Red)  18 (Black) 25 (Black)
  

• Insert 40: 40 is greater than 25, hence it becomes a red child:

       8 (Black)
       /    \
     5 (Black)  17 (Red)
    /         /   \
  1 (Red)  18 (Black) 25 (Black)
                 \
                 40 (Red)
  

• Insert 80: 80 is greater than 40, hence it becomes a red child of 40:

       8 (Black)
       /    \
     5 (Black)  17 (Red)
    /         /   \
  1 (Red)  18 (Black) 25 (Black)
                 \
                 40 (Black)
                        \
                        80 (Red)
  

4. Verify and Summarize

1. We maintained red-black properties during each insertion.
2. We ensured that no two red nodes are adjacent.
3. The heights are balanced.

Final Answer

The final red-black tree structure is:

       8 (Black)
       /    \
     5 (Black)  17 (Red)
    /         /   \
  1 (Red)  18 (Black) 25 (Black)
                 \
                 40 (Black)
                        \
                        80 (Red)