The buddy system in memory allocation is an efficient method designed to reduce fragmentation while facilitating the allocation and deallocation of memory blocks. This system organizes memory into partitions, referred to as ‘buddies’, which are always of sizes that are powers of two, specifically $2^n$, where $n$ is a non-negative integer. This design enables efficient splitting and merging of memory blocks.

The process begins with a single block of memory of size $2^m$. Upon receiving a memory allocation request, the system first checks whether the current block size exceeds the requested size. If it does, the block is divided into two equal ‘buddy’ blocks, each having a size of half the original block. This division continues until the block size is either equal to or slightly larger than the requested size. The system then allocates the smallest block that can adequately accommodate the requested data.

When a block is deallocated, the system examines whether its buddy block is also free. If it is, the two blocks are merged to form a larger block. This merging process continues as long as there are free buddy blocks available to combine. This strategy helps reduce external fragmentation by creating larger blocks of memory when necessary.

The efficiency of the buddy system arises from the fact that all operations—allocation, deallocation, splitting, and merging—can be executed in constant time. Additionally, it optimally utilizes memory by minimizing wasted space. However, the system may experience internal fragmentation if the size of the requested memory is significantly smaller than the allocated block size. For instance, if a process requests $100$ bytes and the smallest available block is $128$ bytes, then $28$ bytes will remain unused.

In conclusion, the buddy system is a dynamic memory allocation scheme that employs a binary tree structure to represent both used and free memory blocks. While it is efficient and minimizes waste, it can be prone to internal fragmentation.