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A B-tree of order 4 and of height 3 will have a maximum of _______ keys.
- a)255
- b)63
- c)127
- d)188
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
A B-tree of order 4 and of height 3 will have a maximum of _______ key...
A B-tree is a self-balancing search tree data structure that maintains sorted data and allows efficient operations such as insertion, deletion, and search. It is commonly used in databases and file systems.
A B-tree of order 4 means that each node in the tree can have a maximum of 4 children. The order of a B-tree refers to the maximum number of children a node can have.
The height of a B-tree is the number of levels in the tree, starting from the root node. In this case, the height of the B-tree is given as 3.
To determine the maximum number of keys in the B-tree, we need to consider the maximum number of nodes at each level and the maximum number of keys in each node.
At the root level (level 0), there can be a maximum of 1 node with 3 keys (4 children - 1). So, at level 0, there can be a maximum of 3 keys.
At level 1, each node can have a maximum of 4 children and a minimum of 2 children. Therefore, the maximum number of nodes at level 1 is 1 (root node) + 4 (maximum number of children at level 1) = 5. Each node at level 1 can have a maximum of 3 keys. Therefore, at level 1, there can be a maximum of 5 nodes * 3 keys = 15 keys.
Similarly, at level 2, each node can have a maximum of 4 children and a minimum of 2 children. Therefore, the maximum number of nodes at level 2 is 5 (maximum number of nodes at level 1) * 4 (maximum number of children at level 2) = 20. Each node at level 2 can have a maximum of 3 keys. Therefore, at level 2, there can be a maximum of 20 nodes * 3 keys = 60 keys.
Adding up the keys at each level, we have 3 + 15 + 60 = 78 keys.
However, this calculation only considers the maximum number of keys in the internal nodes of the B-tree. The leaf nodes, which contain the actual data, will also have keys. Since the B-tree is self-balancing, the number of keys in the leaf nodes will be the same as the number of keys in the internal nodes.
Therefore, the maximum number of keys in a B-tree of order 4 and height 3 is 78 keys * 2 (considering both internal and leaf nodes) = 156 keys.
However, none of the given options match this result. Therefore, it seems there may be an error in the given options or the expected answer.
A B-tree of order 4 means that each node in the tree can have a maximum of 4 children. The order of a B-tree refers to the maximum number of children a node can have.
The height of a B-tree is the number of levels in the tree, starting from the root node. In this case, the height of the B-tree is given as 3.
To determine the maximum number of keys in the B-tree, we need to consider the maximum number of nodes at each level and the maximum number of keys in each node.
At the root level (level 0), there can be a maximum of 1 node with 3 keys (4 children - 1). So, at level 0, there can be a maximum of 3 keys.
At level 1, each node can have a maximum of 4 children and a minimum of 2 children. Therefore, the maximum number of nodes at level 1 is 1 (root node) + 4 (maximum number of children at level 1) = 5. Each node at level 1 can have a maximum of 3 keys. Therefore, at level 1, there can be a maximum of 5 nodes * 3 keys = 15 keys.
Similarly, at level 2, each node can have a maximum of 4 children and a minimum of 2 children. Therefore, the maximum number of nodes at level 2 is 5 (maximum number of nodes at level 1) * 4 (maximum number of children at level 2) = 20. Each node at level 2 can have a maximum of 3 keys. Therefore, at level 2, there can be a maximum of 20 nodes * 3 keys = 60 keys.
Adding up the keys at each level, we have 3 + 15 + 60 = 78 keys.
However, this calculation only considers the maximum number of keys in the internal nodes of the B-tree. The leaf nodes, which contain the actual data, will also have keys. Since the B-tree is self-balancing, the number of keys in the leaf nodes will be the same as the number of keys in the internal nodes.
Therefore, the maximum number of keys in a B-tree of order 4 and height 3 is 78 keys * 2 (considering both internal and leaf nodes) = 156 keys.
However, none of the given options match this result. Therefore, it seems there may be an error in the given options or the expected answer.
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A B-tree of order 4 and of height 3 will have a maximum of _______ key...
A B-tree of order m of height h will have the maximum number of keys when all nodes are completely filled. So, the B-tree will have n = (mh+1 – 1) keys in this situation. So, required number of maximum keys = 43+1 – 1 = 256 – 1 = 255.
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