CONTINUED FRACTION AND BINARY TREE GRAPHS
A continued fraction is a way of representing a real number by a sequence of integers. In this paper, we display an explanation from the continued fraction expansion in a more general state, and we present a new method to think about these continued fractions using tree graphs. Continued fractions, binary tree graphs, the topological index Z, and the Euclidean division algorithm are combined. In fact, we found a new combinatorial realization of the continued fractions with the binary trees and number of connected components of binary trees. Our aim is to show how this realization reflects the convergence of the continued fractions, the topological index Z, and as well as the Euclidean division algorithm. We think that this different perspective can be useful because the continued fraction depends on the order of vertices, which are the set of all positive rational numbers. Thus, the choice of the right sequence for vertices of binary tree has a significant impact on the build of continued fraction. The connection between binary tree, sub binary trees, and continued fractions will be explored. Findings are to establish results on sums of vertices, palindromic continued fractions.