In how many different contexts do the Catalan numbers appear?
Here is a short list of some of the better known (taken from
a paper by Roger Eggleton and Richard Guy in
Mathematics Magazine,
October 1988:

- The number of ways of parenthesizing a (non-associative)
product of n+1 factors.
- The quotient when the middle binomial coefficient (2n!)/(n!)^2
is divided by n+1.
- The number of ways of chopping an (irregular) (n+2)-gon into
n triangles by n-1 non-intersecting diagonals.
- The self-convolving sequence, c[0]=1,

c[n+1] = c[0]c[n] +
c[1]c[n-1] + ... + c[n]c[0]
- The number of (plane) binary trees with n
internal (trivalent) nodes.
- The number of plane rooted trees with n edges.
- The coefficients in the power series expansion of the
generating function (1-sqrt(1-4x))/2x.
- The number of mountain ranges you can draw, using n upstrokes
and n downstrokes.
- The number of single mountains (cols (?) between peaks are
no longer allowed to be as low as sea level) you can draw
with n+1 upstrokes and n+1 downstrokes.
- The number of paths from (0,0) to (n,n) (resp. (n+1,n+1))
increasing just one coordinate by one at each step, without
crossing (resp. meeting) the diagonal x=y at any point
between (a thin disguise of 8 and 9).
- Another disguise is the number of ways n votes can come
in for each of two candidates A and B in an election, with
A never behind B.
- The recurring sequence c[0]=1, (n+2)c[n+1]
= (4n+2)c[n],
(n>=0).
- The number of different frieze patterns with n+1 rows
(see Conway & Coxeter, Math Gaz. 57, 1973).
- The number of ways 2n people, seated round a table, can
shake hands in n pairs, without their arms crossing.
- The number of Murasaki diagrams of n colored incense
sticks which do not involve crossings.

Eggleton and Guy then added another interpretation.
If you randomly select the n+1 values v0, v1,...vn
from the interval [0,1], what is
the probability that the piece-wise linear function

is

convex? A function is convex provided any three of its values
vi,vj,vk (i < j < k) satisfy the inequality

(vi-vj)/(i-j) <= (vj-vk)/(j-k)

E&G show that the answer is c[n]/(n!)^2.

In addition, I've noticed that if A[2k] denotes the average (2k)th
power of distance between two randomly chosen points inside a
spherical ball, then A[2k] = c[k+1]/(k+1).

So here we have 17 different interpretations of the Catalan numbers.
Sloane's Encyclopedia of Integer Sequences says there are "dozens"
of interpretations. Can anyone supply any others?

*Source: http://www.mathpages.com/*