The document outlines the design of a specialized tree data structure with a specific branching factor for nodes based on their parent nodes. It defines the abstract tree type along with necessary functions and axioms using C++. Additionally, it discusses the mathematical perspective of trees as ordered structures that can carry values at each node.

1. Design a special tree data structure such that the branching factor of the root is 3 and the
branching factor of its children is two. After that the branching factor of a node is 2 if the node is
child of a node with branching factor of 3 and its branching factor is 3 if it is the child of a node
with branching factor of 2 in C++.
Solution
As an ADT, the abstract tree type T with values of some type E is defined, using the abstract
forest type F (list of trees), by the functions:
value: T E
children: T F
nil: () F
node: E × F T
with the axioms:
value(node( e , f )) = e
children(node( e , f )) = f
In terms of type theory, a tree is an inductive type defined by the constructors nil (empty forest)
and node (tree with root node with given value and children).
Mathematical[edit]
Viewed as a whole, a tree data structure is an ordered tree, generally with values attached to each
node. Concretely, it is (if required to be non-empty):