WIKI :: A formal system is used for inferring theorems from axioms according to a set of rules. Each formal system uses primitive symbols (which collectively form an alphabet) to finitely construct a formal language from a set of axiomst through inferential rules of formation. The system thus consists of valid formulas built up through finite combinations of the primitive symbols—combinations that are formed from the axioms in accordance with the stated rules.

Every time You boot Your computer (or a smartphone), a new tree emerges in the realm of information.

"create new directory" = establish a new branch = create information

Taxonomy is the practice and science of categorization based on discrete sets. The word is also used as a count noun: a taxonomy, or taxonomic scheme, is a particular categorisation. The word finds its roots in the Greek language τάξις, taxis (meaning 'order', 'arrangement') and νόμος, nomos ('law' or 'science'). Originally, taxonomy referred only to the categorisation of organisms or a particular categorisation of organisms. In a wider, more general sense, it may refer to a categorisation of things or concepts, as well as to the principles underlying such a categorisation. (https://en.wikipedia.org/wiki/Taxonomy_(general) )

From a graph theory perspective, binary (and K-ary) trees as defined here are actually arborescences. A binary tree may thus be also called a bifurcating arborescence—a term which appears in some very old programming books, before the modern computer science terminology prevailed. It is also possible to interpret a binary tree as an undirected, rather than a directed graph, in which case a binary tree is an ordered, rooted tree. Some authors use rooted binary tree instead of binary tree to emphasize the fact that the tree is rooted, but as defined above, a binary tree is always rooted. A binary tree is a special case of an ordered K-ary tree, where k is 2. ( https://en.wikipedia.org/wiki/Binary_tree )