Fixed GA3 again and again

GA3/ga3.tex

@@ -639,7 +639,7 @@ FUNCTION JOIN-INTERVALS(X)
   
   c $\gets$ 1
   n $\gets$ 0
-  start $\gets$ X[i][1]
+  start $\gets$ X[1][1]
   for i from 2 to X.length:
     c $\gets$ c + X[i][2]
     if c == 0:
@@ -654,4 +654,11 @@ FUNCTION JOIN-INTERVALS(X)
 The complexity of this algorithm is $O(n\log(n))$ since the sorting is in $\Theta(n\log(n))$ and the
 union operation afterwards is $\Theta(n)$.
 
+\section{Bonus}
+
+The number of possible trees with $n$ nodes can be defined recursively. The number of trees with 0 or 1 node is clearly 1, since there is no freedom in arranging any remaining elements as children. For $n$ nodes, this number can be defined as the the sum, for each $x, y \in N_0$ s.t. $x + y = n - 1$, of the number of trees with $x$ nodes times the number of trees with $y$ nodes. We consider only $n - 1$ nodes since one node must be the root of the tree. 
+Using math notation, the number $T_n$ of trees with $n$ nodes can be expressed as:
+
+$$T_n = 1 \hspace{1cm} n = 0 \lor n = 1$$
+$$T_n = \sum_{i = 0}^{n-1} C_i C_{n-1-i} \hspace{1cm} n \geq 2$$
 \end{document}