Equivalence of Three Catalan Number Interpretations

n + 2 sides

5

index

1

show edge labels

show tree

The Catalan numbers count:

• the ways a polygon with

n+2

sides can be cut into

n

triangles;

• the ways to parenthesize a sequence of

n+1

factors to be multiplied, two at a time;

• the planted trivalent trees with

n+1

leaves.

This Demonstration shows the equivalence of the three interpretations by optionally labeling each polygon triangulation with the corresponding parenthesized expression, tree, or both.