Self-adjusting binary search trees, introduced by Sleator and Tarjan, are a type of binary search tree that dynamically adjusts its structure based on access patterns. These trees are designed to be efficient in terms of amortized time complexity, making them as effective as balanced trees for total running time. Unlike balanced trees, which rely on explicit structural constraints, splay trees use a restructuring heuristic called splaying. Splaying moves accessed nodes closer to the root, improving future access times. Splay trees have been shown to be as efficient as static optimum search trees for sufficiently long access sequences, and they can adapt to varying access patterns. The efficiency of splay trees comes from their ability to adjust automatically based on usage, without requiring additional storage for balance information. This makes them more space-efficient and easier to implement compared to balanced trees. However, they may require more local adjustments during accesses and can be less efficient in real-time applications. The paper presents an amortized analysis of splay trees, showing that they achieve an amortized time bound of O(log n) per operation. This is achieved through the use of a potential function, which helps in analyzing the overall efficiency of the tree. The analysis also includes various theorems that demonstrate the efficiency of splay trees in different scenarios, such as static optimality, finger search, and working set theorems. The paper also discusses extensions of splaying to other data structures, such as lexicographic or multidimensional search trees and link/cut trees. These extensions simplify the implementation of complex data structures while maintaining the efficiency of splay trees. The paper concludes with a unified theorem that combines the efficiency of splay trees across various operations, showing that they can be as efficient as any dynamically updated binary search tree. The analysis and implementation of splay trees highlight their adaptability and efficiency in handling a wide range of operations, making them a valuable tool in the design of self-adjusting data structures.Self-adjusting binary search trees, introduced by Sleator and Tarjan, are a type of binary search tree that dynamically adjusts its structure based on access patterns. These trees are designed to be efficient in terms of amortized time complexity, making them as effective as balanced trees for total running time. Unlike balanced trees, which rely on explicit structural constraints, splay trees use a restructuring heuristic called splaying. Splaying moves accessed nodes closer to the root, improving future access times. Splay trees have been shown to be as efficient as static optimum search trees for sufficiently long access sequences, and they can adapt to varying access patterns. The efficiency of splay trees comes from their ability to adjust automatically based on usage, without requiring additional storage for balance information. This makes them more space-efficient and easier to implement compared to balanced trees. However, they may require more local adjustments during accesses and can be less efficient in real-time applications. The paper presents an amortized analysis of splay trees, showing that they achieve an amortized time bound of O(log n) per operation. This is achieved through the use of a potential function, which helps in analyzing the overall efficiency of the tree. The analysis also includes various theorems that demonstrate the efficiency of splay trees in different scenarios, such as static optimality, finger search, and working set theorems. The paper also discusses extensions of splaying to other data structures, such as lexicographic or multidimensional search trees and link/cut trees. These extensions simplify the implementation of complex data structures while maintaining the efficiency of splay trees. The paper concludes with a unified theorem that combines the efficiency of splay trees across various operations, showing that they can be as efficient as any dynamically updated binary search tree. The analysis and implementation of splay trees highlight their adaptability and efficiency in handling a wide range of operations, making them a valuable tool in the design of self-adjusting data structures.