The classical Catalan numbers, rediscovered by the Belgian mathematician Eugène Charles Catalan (1814–1894) seem to have been first studied by the famous Swiss mathematician Leonhard Euler (1707–1783). Euler considered the problem of counting the number of ways a given convex polygon1 can be divided into triangles by drawing non-intersecting diagonals (a diagonal is a line segment joining non-adjacent vertices, and two diagonals are considered to be non-intersecting if they intersect only in a vertex of the polygon). Obviously, this number depends only on n, the number of sides of the polygon; obviously, too, (n - 3) diagonals will be drawn, creating (n - 2) triangles. We will call the number2 c n-2, thus putting emphasis on the number of triangles; this accords with standard practice. It is easy to see that, for polygons with 3,4, 5 sides, we have d = 1, c 2 = 2, c 3 = 5, respectively (see Figure 1). Even Euler, however, found it difficult to obtain a general formula for c k .
KeywordsSource Node Binary Tree Good Path Catalan Number Binomial Theorem
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