**Expression trees** are a type of binary tree used to represent expressions in a tree-like structure.

• In an expression tree, each leaf node represents an operand, and each non-leaf node represents an operator.

• The structure of the tree reflects the hierarchical and associative nature of expressions.

• Expression trees naturally capture the associativity of operators. **For example**, in **a+b+c**, the expression tree ensures that the addition is performed from left to right.

**In summary**, expression trees are a powerful representation of mathematical expressions, facilitating their evaluation, manipulation, and optimization.

## Structure of an Expression Tree:

**Operand Nodes**: These are the leaf nodes of the tree and represent the operands (e.g., variables, constants).**Operator Nodes**: These are internal nodes of the tree and represent the operators (e.g., addition, subtraction, multiplication, division).**Parent-Child Relationships**: The operators are the parents of their corresponding operands.

## Applications Expression Trees:

Evaluation of Expressions:

Once an expression is represented as an expression tree, it becomes easier to evaluate the expression by recursively traversing the tree.

Compilation and Interpretation:

Expression trees are used in compilers and interpreters to parse and analyze code.

Mathematical Optimization:

Expression trees can be manipulated to optimize mathematical expressions, such as simplifying algebraic expressions or identifying redundant computations.

Symbolic Mathematics:

They are used in symbolic mathematics systems to perform symbolic manipulation of expressions.

Computer Algebra Systems (CAS):

CAS systems heavily use expression trees for symbolic computation, simplification, and solving equations.

Query Optimization in Databases: In databases, expression trees can be used to represent query plans and optimize query execution.

## Advantages of Expression Trees:

Structured Representation:

Expression trees provide a structured representation of expressions, making it easier to understand and manipulate them.

Efficient Evaluation:

Once constructed, evaluating expressions using expression trees can be efficient, especially for repetitive evaluations.

Ease of Optimization:

The hierarchical structure of expression trees facilitates optimization techniques, such as algebraic simplification and common subexpression elimination.

## Disadvantages of Expression Trees:

Overhead in Construction:

Constructing an expression tree from an expression requires additional computation and memory overhead.

Complexity in Handling Nested Expressions:

Handling deeply nested expressions can lead to complex tree structures, which may increase the complexity of evaluation and manipulation algorithms.

Limited Representation:

Expression trees may not be suitable for representing certain types of expressions, such as those involving recursion or dynamic memory allocation.

## Note:

The expression tree is a binary tree in which each internal node corresponds to the operator and each leaf node corresponds to the operand so for example expression tree for 3 + ((5+9)*2) would be: