Number of valid parenthesis catalan number explanation

While studying about catalan numbers, some of the applications that I came across were:

1. no of possible binary search trees using n nodes.
2. no of ways to draw non-intersecting chords using 2*n points on a circle.
3. no of ways to arrange n pairs of parenthesis.

While I understand the first two problems, how catalan numbers fit in their solution, I am not able to understand how they fit in the third problem.

Couldn't find any other useful resource on the internet which explains the HOW part. Everyone just says that it's the solution.

Can someone please explain.

Since others do not seem to agree with me that this question is off-topic, I now decide that it is on topic and provide and answer.

The Wikipedia is indeed confusing about the "number of ways to arrange n pairs of parentheses" (the second bullet point in this link.) Part of the confusion is that the order of the strings of parentheses does not match the order of the binary tree, which you do understand, or with many of the other examples.

Here is a way to transform a string of n pairs of parentheses which are correctly matched into a binary tree with n internal nodes. Consider the left-most parenthesis, which will be a left-parenthesis, together with its matching right-parenthesis. Turn the string into a node of the binary tree. The sub-string that is inside the currently-considered parentheses becomes the left child of this node, and the sub-string that is after (to the right) of the currently-considered right-parenthesis becomes the right child. Either or both sub-strings may be empty, and the currently-considered parentheses are simply removed. If either sub-string is not empty, continue this procedure recursively until all parentheses have been removed.

Here are two examples. Let's start with the string ((())). We start with

The considered-parentheses are the outermost ones. This becomes

(I did not bother drawing the external leaf nodes) then

then

which is Wikipedia's left-most binary tree with 3 internal nodes.

Now let's do another string, (())(). We start with

Again, the considered-parentheses are the outermost ones. This transforms to

And now the considered-parentheses are the first two, not the outermost ones. This becomes

which finally becomes

which is the second binary tree in Wikipedia's list.

I hope you now understand. Here is a list of all five possible strings of 3 pairs of parentheses that are correctly paired, followed by Wikipedia's list of binary trees. These lists now correspond to each other.

    ((()))       (()())        (())()       ()(())       ()()()