Binary Tree

Trees are mathematical objects in the field of combinatorics, the area of maths about counting things. A full binary tree is a kind of mathematical structure called a graph where nodes (dots) are connected by edges (lines). The full binary tree starts with a root nodes which branches out to the left and right. Each node thereafter either has two more nodes coming from it (called children) or the node is a dead end (called a leaf). If the number of non-leaf nodes in a full binary tree is n, then the total possible number of distinct trees of that same size is given by the nth Catalan number. 

This embroidery design shows the structure of a binary tree with 8 levels of branches. This tree has 127 total internal nodes so the total possible number of trees with this same size is the 127th Catalan number, which is on the order of 1073! 

Can you figure out the first 4 Catalan numbers by drawing full binary trees?