Binary Search Tree in Compiler Design

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Sequential search is the easiest method to search a lexical table. However, as programs grow in size, we need for faster and more efficient ways to search and store data. Sequential search becomes too slow for large lexical tables, so we go for another way where binary search trees (binary search trees) are used.

By organizing data in a hierarchical structure, binary search trees significantly reduce the search time compared to linear methods. In this chapter, we will see how binary search trees work in compiler design.

What is a Binary Search Tree?

Binary search trees are tree structures based on binary tree. A binary search tree is a tree-like data structure where each node represents a token. The tree follows two simple rules:

• All nodes in the left subtree of a node contain values smaller than the node’s value.
• All nodes in the right subtree of a node contain values larger than the node’s value.

This structure is used for efficient searching. We can use this structure for insertion and deletion of tokens as well.

Why are Binary Search Trees Used in Compiler Design?

In compiler design, the binary search trees are used to implement symbol tables or lexical tables. These tables store information about variables, functions, and other tokens in a program. This helps in faster searching. Compared to sequential search, binary search trees offer a more efficient way to find tokens. This helps more when the size of the table grows.

For example, if a program has hundreds of variables, a binary search tree gives the compiler option to locate a specific variable much faster than scanning through a list.

How Do Binary Search Trees Work

When adding a token to a binary search tree −

• Start at the Root − Begin the search at the root of the tree.
• Compare Values − Compare the token's value to the value of the current node:
  • If the token is smaller, move to the left subtree.
  • If the token is larger, move to the right subtree.
• Insert the Token: When we reach an empty spot, then insert the token as a new node.

To find a token, the process is similar. We will start at the root, compare values, and move left or right based on the comparison.

Example of Building a Binary Search Tree

Let us see an example for a better understanding. To construct a binary search tree step by step using the tokens frog, tree, hill, bird, bat, and cat.

Step 1: Add the First Token

The first token, frog, becomes the root of the tree.

Step 2: Add tree

Tree is larger than frog, so it becomes the right child.

Step 3: Add hill

Step 4: Add bird

Bird is smaller than frog and smaller than hill, so it is added to the left of hill.

Step 5: Add bat

Bat is smaller than frog, smaller than hill, and smaller than bird, so it becomes the left child of bird.

Step 6: Add cat

Cat is smaller than frog, smaller than hill, and larger than bird, so it becomes the right child of bird.

With that, the binary search tree is complete.

Searching in a Binary Search Tree

To search for a token in a binary search tree, we can start at the root and follow the comparison rules.

Example: Searching for Cat

• Compare cat with the root (frog). The cat is smaller. Then move to the left subtree.
• Compare cat with hill. The cat is smaller, move to the left subtree again.
• Compare cat with bird. The cat is larger, then move to the right subtree.
• Found cat

This process is taking only four comparisons. Even though the tree contains six tokens.

Advantages of Binary Search Trees

Binary search trees offer several benefits. It is especially for large programs −

• Faster Search − In a balanced tree, search time is proportional to the height of the tree (O(log n)). This is much faster than sequential search (O(n)).
• Efficient Insertion and Deletion − Adding or removing tokens is simple. This is provided the tree remains balanced.
• Dynamic Growth − The binary search trees can grow or shrink as needed. Unlike fixed-size arrays.

Limitations of Binary Search Trees

Like other methods this also have limitations.

• Unbalanced Trees − If tokens are added in a sorted order, the tree can become unbalanced and degrade into a linked list (O(n) performance). Then AVL trees will be a better choice.
• Complexity − Maintaining a balanced tree requires additional effort. Especially for self-balancing binary search trees like AVL trees or Red-Black trees.
• Memory Overhead − Each node requires extra memory to store pointers to its left and right children.

Real-World Application of Binary Search Trees

Binary search trees are widely used in −

• Symbol Tables − To store information about variables, functions, and classes in programming languages.
• Databases − For indexing and searching records efficiently.
• Search Engines − To quickly locate and rank search results.

For example, when a compiler encounters a variable in a program then it uses the symbol table (implemented as a binary search tree) to find the variable's details, like its type and scope.

Alternatives to Binary Search Trees

If the limitations of binary search trees are a concern, the other data structures might be more suitable −

• Hash Tables − This offer faster lookups (O(1) in most cases) but lack the hierarchical structure of binary search trees.
• Self-Balancing Trees − Like AVL trees or Red-Black trees, this gives that the tree remains balanced for optimal performance.

We will have a detailed discussion on hash tables in the next chapter.

Conclusion

In this chapter, we explained the concept of binary search trees and their role in compiler design. Starting with the basics, we explored how binary search trees organize data hierarchically and allow for faster searches compared to sequential methods.

Through an example, we saw how tokens are added to a binary search tree and how the tree structure makes searching efficient. We also discussed the advantages and limitations of binary search trees and highlighted their real-world applications.