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Algebraic Expressions:

An algebraic expression is a mathematical phrase that contains numbers, variables, and mathematical operations. These expressions can be simple, like “2x + 3,” or complex, involving multiple terms and operations. Algebraic expressions are commonly used in mathematics to represent relationships and solve equations.

Some common components of algebraic expressions include:

  • Variables: Symbols that represent unknown or changing values, typically denoted by letters like x, y, or z.
  • Constants: Fixed values that do not change, such as numbers like 2, 3, or -5.
  • Operators: Symbols representing mathematical operations like addition (+), subtraction (-), multiplication (*), division (/), exponentiation (^), etc.
  • Terms: Parts of an expression separated by operators. For instance, in the expression “3x + 2y – 5,” “3x,” “2y,” and “-5” are individual terms.

Complete Binary Tree:

A complete binary tree is a special type of binary tree where all levels are completely filled except possibly for the last level, which is filled from left to right. This means that all nodes are as far left as possible at each depth level.

In a complete binary tree:

  • All levels, except possibly the last one, are fully filled with nodes.
  • If the last level is not completely filled, nodes are added from left to right.
  • No gaps are allowed except at the right end of the last level.

Example of a complete binary tree:


/ \

2 3

/ \

4 5

Extended Binary Trees:

An extended binary tree (also known as a 2-tree) is a binary tree that may have additional nodes called external nodes or dummy nodes added to ensure that every internal node has exactly two children. These external nodes serve as placeholders and do not contain any data.

The purpose of extended binary trees is to simplify certain algorithms or data structures that require a fixed number of children for each node.

Example of an extended binary tree:


/ \

2 3

/ \ / \

4 5 6 7

In this example, the nodes 6 and 7 are external nodes, added to ensure that each internal node has exactly two children.