Which of the following ensures minimal disk accesses in B-tree searches ?

Concept: A B-tree is a self-balancing, disk-oriented multiway search tree. Because a random disk read is far slower than any CPU work, B-tree design is optimized to minimize the NUMBER OF DISK BLOCKS READ during a search, which equals the tree's height. Search cost is O(log_t n), where t (the minimum degree, i.e. the fan-out/order) controls how many keys and children each node holds.

Each B-tree node is allocated a maximum capacity (a maximum number of keys and child pointers) chosen to match the size of one disk block — this maximum capacity is the LARGE FAN-OUT (high order). Because a single disk read can therefore examine many keys at once, height stays low: even in the worst case, where every node holds only the guaranteed minimum share of that capacity, the tree's height still grows merely logarithmically, with a large base tied to the order. This bounded, shallow height directly translates into very few disk accesses per search — which is exactly why a large fan-out (order) is the design property that ensures minimal disk accesses, regardless of how full any individual node happens to be.

• Root splitting is a maintenance step performed during insertion when a node overflows its capacity, so it keeps the tree balanced at write time — it is not itself a property that reduces disk accesses during a search.

• Variable-sized nodes are not how B-trees are built: nodes follow a fixed order/degree bound matched one-to-one with a disk block; letting node size vary unpredictably would break that mapping to disk blocks.

• Minimum height for a fixed number of keys is the smallest possible number of levels, achievable when every node is packed to its maximum capacity at every level. A B-tree does not guarantee this: it only guarantees that each non-root node stays at least half full, so a valid but under-filled B-tree can need more levels than that theoretical minimum. Its height is still bounded and grows logarithmically because of the order, but bounded-and-logarithmic is not the same guarantee as strictly minimal — the actual design lever is the large fan-out (order), not an independent 'minimum height' property.

So the design choice engineers actually make — a large fan-out (order) — is what ensures minimal disk accesses in B-tree searches.